(A) (12, 5) (B) ( 8, 15) (C) (3,6) (D) (4,4)

Transcription

1 DR. YOU: 018 FALL 1 CHAPTER 1. ANGLES AND BASIC TRIG LECTURE 1-0 REVIEW EXAMPLE 1 YOUR TURN 1 Simplify the radical expression. Simplify the radical expression. (A) 108 (A) 50 First, find the biggest perfect square which divides the radicand 108; 108 = 36 3 = 36 3 (B) 7 = 6 3 (B) 3 First, find the biggest perfect square which divides the radicand 3; 3 = 16 (C) 48 = 16 = 4 (C) (3 8) (4 1) (3 8) (4 1) = (D) (5 6) 18 = = 48 6 (D) ( 5 3)( 5 + 3) = : FOIL (E) (3 + 4)(3 4) = = 11

2 DR. YOU: 018 FALL EXAMPLE YOUR TURN Find the distance between ( 3,4) and the origin. Find the distance between the given point and the origin. (A) (1, 5) The distance is D = (x x 1 ) + (y y 1 ) (4 0) + ( 3 0) = = 5 = 5 (B) ( 8, 15) (C) (3,6) (D) (4,4)

3 DR. YOU: 018 FALL 3 EXAMPLE 3 YOUR TURN 3 Simplify the following number. 7π 6 π 3 Simplify the following number. (A) 11π 1 3π 4 7π 6 π 3 = π ( ) Factor out π = π ( 1 ) Subtract fractions = π (B) 5π 4 π (C) π 3π 6

4 DR. YOU: 018 FALL 4 EXAMPLE 4 YOUR TURN 4 Simplify the following number. (A) π/6 /3 Simplify the following number. (A) 15 π 180 π 6 3 = 1 6 π 3 = ( ) π = ( 1 4 ) π = π 4 (B) π/3 π/ π 3 π = 3 π 1 π = 3 1 = 4 3 (B) 3π 4 5π 6 (C) 5π

5 DR. YOU: 018 FALL 5 EXAMPLE 5 YOUR TURN 5 Rationalize the denominator: 5 3 Rationalize the denominator: (A) = ( 3)( 3) = = (B) 3 5 EXAMPLE 6 YOUR TURN 6 Rationalize the denominator: 5 10 Rationalize the denominator: (5 + 10) = (5 10)(5 + 10) (5 + 10) = = = (5 + 10)

6 DR. YOU: 018 FALL 6 EXAMPLE 7 YOUR TURN 7 Simplify the expression Simplify the expression (A) = = (1 3 ) = = (B) 1 3

7 DR. YOU: 018 FALL 7 LECTURE 1-1 ANGLES AND THEIR MEASURE When we work with objects in mathematics it is convenient to give them names. These names are arbitrary and can be chosen to best suit the situation or mood. For example, if we are in a romantic mood we could use, or any number of other symbols. Traditionally in mathematics we use letters from the Greek alphabet to denote angles. This is because the Greeks were the first to study geometry. The Greek alphabet is shown below (do not worry about memorizing this). α- alpha β-beta γ-gamma δ-delta ε-epsilon ζ-zeta η-eta θ-theta ι-iota κ-kappa λ-lambda μ-mu ν-nu ξ-xi ο-omicron π-pi ρ-rho σ-sigma τ-tau υ-upsilon φ-phi χ-chi ψ-psi ω-omega We will use Pythagorean theorem which is named after the Greek philosopher Pythagoras, though it was known well before his time in different parts of the world such as the Middle East and China. ANGLE MEASURE 1) There are two ways to measure an angle: degree (one counterclockwise revolution = 360 ) and radian π radian = radian = 180 { π = π radian radian 180 ) One degree 1 = 60 (60 minutes). One minute 1 = 60 (60 seconds) 1 radian r r 3) When we need to be convert between degrees and radians, let θ be the measure of the angle in degrees and θ R the measure of the angle in radians θ 180 = θ R π Degrees 180 = Radians π

8 DR. YOU: 018 FALL 8 EXAMPLE 1 YOUR TURN 1 Convert 155 to radians. Convert each angle in degrees to radians. (A) π = = π = π (B) 70 = 31π 36 rad (C) 1050 (D) 540

9 DR. YOU: 018 FALL 9 EXAMPLE YOUR TURN Convert 4π 3 to degrees. Convert each angle in radians to degrees. (A) π 4 4π 3 = 4π π = = (B) 5π 1 = 40 (C) 11π 6 (D) 7π

10 DR. YOU: 018 FALL 10 EXAMPLE 3 YOUR TURN 3 Convert radians to degrees. (A) Convert 6 radians to degrees. = 180 π = ( 360 π ) (B) Convert 3 radians to degrees. (C) Convert 5 radians to degrees.

11 DR. YOU: 018 FALL 11 STANDARD POSITION: If we position an angle in the xy-plane with the vertex is at the origin and one side along the positive x-axis, then the angle is said to be in standard position. The side that lies along the positive x-axis is called the initial side. The other side opens counter-clockwise and is called the terminal side. y y terminal side θ is negative in standard position x θ intial side x θ intial side θ is postive in standard position terminal side EXAMPLE 4 YOUR TURN 4 Draw 10 in standard position. Draw 10 in standard position. 10 EXAMPLE 5 YOUR TURN 5 Draw 135 in standard position Draw 150 in standard position -135

12 DR. YOU: 018 FALL 1 EXAMPLE 6 YOUR TURN 6 Draw 5π 6 in standard position. Draw 4π 3 in standard position. 5π 6 =150 EXAMPLE 7 YOUR TURN 7 Draw 410 in standard position. Draw 570 in standard position. 410 EXAMPLE 8 YOUR TURN 8 Draw 550 in standard position. Draw 460 in standard position. 550 = ( 190 )

13 DR. YOU: 018 FALL 13 EXAMPLE 9 YOUR TURN 9 Convert to a decimal in degrees. Round the Convert to a decimal in degrees. Round the answer to four decimal places. answer to four decimal places = = ( 1 60 ) + 17 ( ) = EXAMPLE 10 YOUR TURN 10 Convert to a decimal in degrees. Round the Convert to a decimal in degrees. Round the answer to four decimal places. answer to four decimal places = = ( 1 60 ) + 0 ( ) =

14 DR. YOU: 018 FALL 14 EXAMPLE 11 YOUR TURN 11 Convert in the D M S form. Round the answer Convert in the D M S form. Round the answer to the nearest second. to the nearest second = (60 ) = = = (60 ) = = EXAMPLE 1 YOUR TURN 1 Convert in the D M S form. Round the answer to Convert 5.56 in the D M S form. Round the answer to the nearest second. the nearest second = (60 ) = = = (60 ) = =

15 DR. YOU: 018 FALL 15 NOTE: Coterminal angles are angles in standard position (angles with the initial side on the positive xx -axis) that have a common terminal side. EXAMPLE 13 YOUR TURN 13 Find the angle θ in 0 θ < π such that its terminal side Find the angle θ in 0 θ < π such that its terminal side are same to the terminal side of 800 are same to the terminal side of (A) ) 800 is a positive angle such that is bigger than 360 ) To find the coterminal angle in 0 θ < π, we keep subtracting 360 until we get an angle which is in 0 θ < π. 800 (360 ) = 80 (B) 780 3) Then, 800, 80 have the same terminal side. θ = 80 (C) 190 EXAMPLE 14 YOUR TURN 14 Find the angle θ in 0 θ < π such that its terminal side Find the angle θ in 0 θ < π such that its terminal side are same to the terminal side of 800 are same to the terminal side of (A) 860 1) 800 is a negative angle. ) To find the coterminal angle in 0 θ < π, we keep adding 360 until we get an angle which is in 0 θ < π (360 ) = 80 (B) 150 (C) 345 1) Then, 800, 80 have the same terminal side. θ = 80

16 DR. YOU: 018 FALL 16 EXAMPLE 15 YOUR TURN 15 Find the angle θ in 0 θ < π such that its terminal side are same to the terminal side of 49π 1 Find the angle θ in 0 θ < π such that its terminal side are same to the terminal side of (A) 14π 6 1) 49π 1 is a negative angle ) To find the coterminal angle in 0 θ < π, we keep adding π until we get an angle which is in 0 θ < π. 49π 1 + 3(π) = 3 1 π 3) Then, 49π, θ = 3 1 π π have the same terminal side. (B) 5π 4 (C) 3π 3

17 DR. YOU: 018 FALL 17 A reference angle for a given angle in standard position is the positive acute angle formed by the x-axis and the terminal side of the given angle. EXAMPLE 16 YOUR TURN 16 Find the reference angle for the following angle Find the reference angle for the following angle (A) 0 (A) 40 1) First, the angle φ in 0 φ < π such that its terminal side are same to the terminal side of 0 φ = 0 is in III quadrant ) Find the difference between 180 and = 40 The reference angle is 40 (B) 13π 3 1) First, the angle φ in 0 φ < π such that its terminal side are same to the terminal side of 13π 3 φ = 5π 3 = 13π 3 + 6π φ = 5π 3 is in IV quadrant (B) 5π 6 (C) 15π 4 ) Find the difference between π and 5π 3 π 5π 3 = π 3 The reference angle is π 3

18 DR. YOU: 018 FALL 18 HOMEWORK 1 NAME As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to this coversheet. 1. Draw the angle 135 in the standard position. Draw the angle 3π in the standard position 3. Convert the following degrees in radians (A) 65 (B) Convert the following radians in degrees. (A) 5π 6 (B) 4π 3 (C) (D) 3 5. Convert the angle to a decimal in degrees. Round the answer to four decimal places. 6. Convert the angle in the D M S form. Round the answer to the nearest second. 7. Find a coterminal angle θ in 0 θ < π of each following angle. (A) 1170 (B) 980 (C) 11π 3 (D) 53π 1 8. Find the reference angle of each following angle. (A) 165 (B) 50

19 DR. YOU: 018 FALL 19 LECTURE 1- ARC LENGTH AND SECTOR AREA; LINEAR AND ANGULAR SPEED AREA OF SECTOR: angle θ is radians angle θ is degrees Length of arc s = rθ s = θ 360 πr The sector area A = 1 r θ A = θ 360 r π center O A : area of sector r: radius s: arc length EXAMPLE 1 YOUR TURN 1 Find the length of the arc and the area of the sector that (A) Find the length of the arc and the area of the subtends a central angle of 60 using a radius of 9 meters. sector that subtends a central angle of 1 radian using a radius of 10 meters. 1) Convert the angle in degree to radian. π = π 3 ) Use the formula to find s, and A The arc length is s = rθ = 9 π 3 = 3π The area of the sector is A = 1 r θ = 1 9 π 3 = 7π (B) Find the length of the arc and the area of the sector that subtends a central angle of 30 using a radius of 5 ft. Therefore, Arc length is 3π 7π m, and area of the sector is m

20 DR. YOU: 018 FALL 0 EXAMPLE YOUR TURN Find the radius r and the area of a sector whose central Find the central angle θ in radian and the area of a sector angle is θ = 30 and the arc length is s = 1 feet. whose radius is r = 0 feet and the arc length is s = 16 feet. 1) Convert the angle in degree in radian. θ = 30 = π 6 radian ) First, find the radius by using s = 1 and θ = π 6 s = rθ 1 = r π π = r r = 7 π 3) Find the area of the sector A = 1 r θ = 1 (7 π ) π 6 = 43π Therefore, the radius is 7 π ft and the area of the sector is 43π ft

21 DR. YOU: 018 FALL 1 EXAMPLE 3 YOUR TURN 3 Find the radius r and the arc length of the following sector (A) Find the radius r and the arc length of a sector whose central angle is θ = 1 radian and the area of the whose central angle is θ = 4 radian and the area of the sector is A = 3 square feet. sector is A = 4 square feet. 4) Convert the angle in degree in radian. θ = 1 radian 5) First, find the radius by using A = 4 and θ = 1 A = 1 r θ 4 = 1 r 1 4 = 1 4 r 16 = r 16 = r since r is radian (r > 0) r = 4 (B) A wedge-shaped slice of pizza has an area of 60 cm. The end of the slice makes an angle of 35. What was the radius of the pizza from which the slice was taken? 6) Find the arc length s = rθ = 4 1 = Therefore, the radius is 4 ft and arc length is ft

22 DR. YOU: 018 FALL EXAMPLE 4 YOUR TURN 4 The arm and blade of a windshield wiper have a total length of 34 inches. If the blade is 5 inches long and the wiper sweeps out an angle of 150, how much window area can the blade clean? Find the shaded area in a sector; cm 4 cm The area which the blade clean is ) Change θ in radian π 150 = = 5 6 π ) Therefore, the area is Big sector area small sector area = π 1 (34 5) 5 6 π = π 3 4 π = 5375 π in 1

23 DR. YOU: 018 FALL 3 SPEED: The angle θ is measured in radian and r is radius Angular speed: ω = π the number of rotations per unit time = angle θ time t Linear speed: v = total distance = rθ time t = circumference the number of rotations per unit time = rω NOTE: Find angular speed of an object which moves along a circular path with radius 10. Number of Rotations per minute Angular speed Linear speed 1 rotation rotations 4 rotations 11 rotations EXAMPLE 5 YOUR TURN 5 A wheel is rotating at 100 revolutions per second. Find the angular velocity in radians per second. (A) A wheel is rotating at 150 revolutions per second. Find the angular velocity in radians per second # of Rotations per second Total angle measure 100 rotations 100 π = 00π rad Angular speed = = total angle measure time 00π rad 1 sec (B) A wheel is rotating at 300 revolutions per second. Find the angular velocity in radians per second = 00π rad/sec

24 DR. YOU: 018 FALL 4 EXAMPLE 6 YOUR TURN 6 An object sweeps out a central angle of π radians in seconds as it moves along a circle of radius 3 m. Find its linear and angular speed over that time period. An object is traveling around a circle with a radius of meters. If the object is released and travels 5 meters in 0 seconds, what is its linear speed? What is its angular speed? In this case, we know total measure of angle: π 3 radians Angular speed total measure of angle = time = π 3 1 = π 3 rad/sec Linear speed = (angular speed) r = π 3 3 = π m/sec EXAMPLE 7 YOUR TURN 7 A child is spinning a rock at the end of a -foot rope at the rate of 180 revolutions per minute (rpm). Find the linear speed of the rock when it is released. A child is spinning a rock at the end of a 6-foot rope at the rate of 10 revolutions per minute (rpm). Find the linear speed of the rock when it is released. We do not know the total measure of angle, but we know the number of revolutions per minute: 180 Total measure of angle: = (number of rotations) π = 180 π = 360π rad Angular speed = total measure of angle time = 360π rad 1 sec = 360π rad/sec Linear speed = (angular speed) r = 360π = 70π ft/sec

25 DR. YOU: 018 FALL 5 EXAMPLE 8 YOUR TURN 8 To approximate the speed of the current of a river, a Cathy is riding a bike with 14-inch diameter wheels circular paddle wheel with diameter 8 feet is lowered into which are rotating 3 revolutions per second. What is her the water. If the current causes the wheel to rotate at a linear speed in feet per minute? (Round to the nearest speed of 10 revolutions per minute, what is the speed of hundredth of a foot per minute, and use the π key on your the current? Express your answer in miles per hour (1 foot calculator to perform the computation) = 1 inches, 1 mile = 580 feet). We know 1) Number of revolution per min: 10 ) Diameter is 8 ft.: radius is 4 ft. Total measure of angle: = (number of rotations) π = 10 π = 0π rad Angular speed = total measure of angle time = 0π rad 1 min = 0π rad/min Linear speed = (angular speed) r = 0π 4 = 80π ft/min But, we change the unit from ft./min to mi/hr. 80π = ft min 80π ft 1 min.8560 mi hr 60 min 1 miles 1 hr 580 ft

26 DR. YOU: 018 FALL 6 EXAMPLE 9 YOUR TURN 9 A bicycle with 0-inch diameter wheels is traveling A bicycle with 34-inch diameter wheels is traveling at 10 mi/h. Find the angular speed of the wheels in radians at 17 mi/h. Find the angular speed of the wheels per minute. How many revolutions per minute do the in radians per minute. How many revolutions per minute wheels make? (Round your answer to three decimal do the wheels make? (Round your answer to three places.) decimal places.) 1) Find the radius in miles r = 0 = 10 inches = = 1 58 miles ) Linear speed = angular speed radius 10 mi/h = ω ω = 580 rad h rad = 1 h 1 h 60 min = 88 rad/min 3) angular speed = π RPM 88 = π RPM RPM = 88 π EXAMPLE 10 YOUR TURN 10 A wheel on a tractor has a 0-inch diameter. How many A wheel on a tractor has a 30-inch diameter. How many full revolutions does the wheel make if the tractor full revolutions does the wheel make if the tractor travels 5 miles? (Round your answer down to the nearest travels 5 miles? (Round your answer down to the nearest whole number.) whole number.) 1) Find the radius in miles r = 0 = 10 inches = miles ) Distance =circumference number of rotations 3 miles = π ( 1 ) number of rotations 58 number of rotations = 3 58 π 5.101

27 DR. YOU: 018 FALL 7 HOMEWORK NAME As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to this coversheet. 1. A wheel is rotating at 50 revolutions per second. Find the angular velocity in radians per second.. An object is traveling around a circle with a radius of 4 centimeters. If in 0 seconds a central angle of 5 radian is 6 swept out, what is the angular speed of the object? What is its linear speed? 3. Find the length of the arc that subtends a central angle of 3π 5 radian using a radius of 5 centimeters. 4. Find the length of the arc that subtends a central angle of 150 using a radius of 5 centimeters. 5. Find the area of a section of a circle of radius 6 cm by subtended the central angle of Find the radius of circle if the area of a section of a circle by subtended the central angle of radian is 18 cm 3 7. A bike tire has a diameter of 36 inches. If the tire rotates 10 revolutions, how far did the bicycle travel? 8. Find the linear speed of a vehicle in feet per minute whose tires have 0 inch diameters and are moving at 5 revolutions per minute. (Round to the nearest hundredth of a foot per minute, and use the π key on your calculator to perform the computation.) 9. Cathy is riding a bike with 16-inch diameter wheels which are rotating 5 revolutions per second. What is her linear speed in feet per minute? (Round to the nearest hundredth of a foot per minute, and use the π key on your calculator to perform the computation.)

28 DR. YOU: 018 FALL 8 LECTURE 1-3 INTRODUCTION TO TRIGONOMETRIES PYTHAGOREAN THEOREM: Given a right triangle with sides of length a, b and c (c being the longest side, which is also called the hypotenuse) then a + b = c. TRIGONOMETRIC FUNCTIONS IN THE RIGHT TRIANGLE sin t = cos t = opposite hypotenus adjacent hypotenus tan t = opposite adjacent csc t = 1 sin t sec t = 1 cos t cot t = 1 tan t hypotenuse t adjacent opposite TIP: SOHCAHTOA (pronounced so-cah-tow-ah) TIP: ELEVATION AND DEPRESSION S sine opp hyp OH C cosine adj hyp AH T tangent opp adj OA EXAMPLE 1 YOUR TURN 1 Use the Pythagorean theorem to find the missing side of Use the Pythagorean theorem to find the missing side of the right triangle shown below. the right triangle shown below Since a = 3 and b = 7, = c 58 = c Since c > 0, c = 58

29 DR. YOU: 018 FALL 9 EXAMPLE YOUR TURN Find sin θ, cos θ, and tan θ in the figure. Find sin θ, cos θ, and tan θ in the figure. θ θ Opposite side is 8 Adjacent side is 15 Hypotenuse is 17 1) sin θ = opposite hypotenus = 8 17 ) cos θ = adjacent = 15 hypotenus 17 3) tan θ = opposite adjacent = 8 15 EXAMPLE 3 YOUR TURN 3 Find sin θ, cos θ, and tan θ in the figure. Find sin θ, cos θ, and tan θ in the figure. 4 θ θ Opposite side is 10 Adjacent side is 4 Hypotenuse is = 116 = 9 1) sin θ = opposite = 10 = 5 = 5 9 hypotenus ) cos θ = adjacent = 4 = = 9 hypotenus ) tan θ = opposite adjacent = 10 4 = 5

30 DR. YOU: 018 FALL 30 EXAMPLE 4 YOUR TURN 4 Use the exact values of appropriate trig functions to Use the exact values of appropriate trig functions to find the sides b and h in the triangle below. find the sides a and b in the triangle below. 1) To find b, tan ( π 3 ) = b 5 3 = b 5 b = 5 3 ) To find h, cos ( π 3 ) = 5 h 1 = 5 h h = 10 EXAMPLE 5 YOUR TURN 5 Use a trig function and your calculator to Use a trig function and your calculator to approximate the value of the unknown sides. approximate the value of the unknown sides. 1) To find a, cos(41 ) = a 63 a = 63 cos(41 ) ) To find b, sin(41 ) = b 63 b = 63 sin(41 )

31 DR. YOU: 018 FALL 31 EXAMPLE 6 YOUR TURN 6 From a point on the ground 47 feet from the foot of a tree, the angle of elevation of the top of the tree is 35º. Find the height of the tree to the nearest foot. From a point on level ground 30 yards from the base of a building, the angle of elevation is 38. Approximate the height of the building to the nearest foot. 1) Draw the figure: ) Find the height x of the tree; tan(35 ) = opposite adjacent = x tan(35 ) = x x = 47 tan(35 ) 33 ft ft EXAMPLE 7 YOUR TURN 7 From point P on the ground, the angle of elevation of an airplane is 3º. The altitude of the plane is 100 meters. What is the distance from point P to the airplane, to the nearest tenth of a meter? The angle of elevation of an airplane from a point P on the ground is 3. If the airplane's altitude is 500 m, how far away is the airplane from P? 1) Draw the figure ) Find the length x of the ladder; sin(3 ) = 100 x x sin(3 ) = 100 x = 100 sin(3 ) m

32 DR. YOU: 018 FALL 3 EXAMPLE 8 YOUR TURN 8 An 8 feet metal guy wire is attached to a broken stop sign A 30-ft ladder leans against a building so that the angle to secure its position until repairs can be made. Attached between the ground and the ladder is 80. How high does to a stake in the ground, the guy wire makes an angle of the ladder reach up the side of the building? 51º with the ground. How far from the foot of the stop sign is the stake, to the nearest tenth of a foot? 1) Draw the figure ) Find the distance from foot of the stop to ground. Then AB = hypotenuse and BC = opposite sin(a) = BC AB sin(51 ) = BC 8 8 sin(51 ) = BC BC = 8 sin(51 ) 6. ft

33 DR. YOU: 018 FALL 33 EXAMPLE 9 YOUR TURN 9 From the top of a light house 60 meters high with its base A blimp 480 ft above the ground measures an angle of at the sea level, the angle of depression from the light depression of 4º from its horizontal line of sight to the house s top to a boat is 15 degrees. What is the distance of base of a house on the ground. Assuming the ground is boat from the foot of the light house? flat, how far away along the ground is the house from the blimp? 1) Draw the figure. Let OA be the height of the light house and B be the position of boat. Since IAB = 15 and IA is parallel to BO, OBA = 15 and OA = 60 m ) Find the distance of OB tan(15 ) = OA OB tan(15 ) = 60 OB OB tan(15 ) = OA OB = 60 tan(15 ) 3.9 m

34 DR. YOU: 018 FALL 34 EXAMPLE 10 YOUR TURN 10 Two students want to determine the heights of two buildings. They stand on the roof of the shorter building. The students use a clinometer to measure the angle of elevation of the top of the taller building. The angle is 44. From the same position, the students measure the angle of depression of the base of the taller building. The A radio tower is located 600 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is 3, and that the angle of depression to the bottom of the tower is 0. How tall is the tower? (Round your answer to four decimal places.) angle is 53 o. The students then measure the horizontal distance between the two buildings. The distance is 18 m. The students drew this diagram. How tall is each building? 1) Draw the figure. In the figure, the length of bigger building is AC + CD ) To find AC tan(44 ) = AC tan(44 ) = AC 3) To find CD tan(53 ) = CD tan(53 ) = b 4) The length of bigger building is AC + CD = 18 tan(44 ) + 18 tan(53 ) 41.9 m

35 DR. YOU: 018 FALL 35 EXAMPLE 11 YOUR TURN 11 Simon bought a new shop and wants to order a new sign for the roof of the building. From point P, he finds the angle of elevation of the roof, from ground level, to be 31º and the angle of elevation of the top of the sign to be 4º. If point P is 4 feet from the building, how tall is the sign to the nearest tenth of a foot? 1) Draw the figure There is an antenna on the top of a building. From a location 600-feet from the base of the building, the angle of elevation to the top of the building is measured to be 35. From the same location, the angle of elevation to the top of the antenna is measured to be 37. Find the height of the antenna. (Round your answer to three decimal places.) ) Find the height a from ground to top of sign tan(4 ) = a 4 4 tan(4 ) = a 3) Find the height b from ground to bottom of sign tan(31 ) = b 4 4 tan(31 ) = b 4) Find the length of sign a b = 4 tan(4 ) 4 tan(31 ) = 7. ft

36 DR. YOU: 018 FALL 36 EXAMPLE 1 YOUR TURN 1 A 400-foot tall monument is located in the distance. From A 400-foot tall monument is located in the distance. From a window in a building, a person determines that the angle a window in a building, a person determines that the angle of elevation to the top of the monument is 15, and that of elevation to the top of the monument is 1, and that the angle of depression to the bottom of the tower is 10. the angle of depression to the bottom of the tower is 6. How far is the building from the monument along the how far away along the ground is the building from the ground? (Round your answer to three decimal places.) monument? (Round your answer to three decimal places.) 1) Draw the figure x a 400 ft b In the figure, the length of tower is a + b = 400 ) To find a tan(15 ) = a x x tan(15 ) = a 3) To find b tan(10 ) = b x x tan(10 ) = b 4) The length of tower is 400 = a + b = x tan(15 ) + x tan(10 ) 400 = x{tan(15 ) + tan(10 )} x = ft tan(15 ) + tan(10 )

37 DR. YOU: 018 FALL 37 EXAMPLE 13 YOUR TURN 13 The angle of elevation of a tower at a point is 45. After going 40 m towards the foot of the tower the angle of elevation of the tower becomes 60. Calculate the height of the tower. A diver stands on a diving board above two swimmers. The angles of depression from the diving board to each swimmer are 30 and 45. If the swimmers are 6 feet apart, then how high is the diving board? 1) Draw the figure. ) Let AB be the height of the tower. Use two right triangles to find x tan(60 ) = AB x AB = x tan(60 ) tan(45 ) = AB AB = (x + 40) tan(45 ) x + 40 Then, x tan(60 ) = (x + 40) tan(45 ) x tan(60 ) = x tan(45 ) + 40 tan(45 ) x tan(60 ) x tan(45 ) = 40 tan(45 ) x(tan(60 ) tan(45 )) = 40 tan(45 ) x = 40 tan(45 ) tan(60 ) tan(45 ) ) Find the AB; AB = x tan(60 ) 94.64

38 DR. YOU: 018 FALL 38 HOMEWORK 3 NAME As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to this coversheet. 1. Find sin θ, cos θ, and tan θ in the figure. (A) 8 17 (B) θ 5 θ 1. A tree casts a shadow of 30 meters when the angle of elevation of the sun is 5. Find the height of the tree to the nearest hundredth of a meter. 3. A -ft ladder leans against a building so that the angle between the ground and the ladder is 63. How high does the ladder reach up the side of the building? 4. From the top of a fire tower, a forest ranger sees his partner on the ground at an angle of depression of 40º. If the tower is 45 feet in height, how far is the partner from the base of the tower, to the nearest tenth of a foot? 5. How long a ladder is needed to reach a windowsill 40-feet above the ground if the ladder rests against the building making an angle of 3π 8 with the ground? Round to the nearest foot. 6. A radio station tower was built in two sections. From a point 87-feet from the base of the tower, the angle of elevation of the top of the first section is 5º, and the angle of elevation of the top of the second section is 40º. To the nearest foot, what is the height of the top section of the tower? 7. (extra) At a certain distance, the angle of elevation to the top of a building is 60º. From 40 feet further back, the angle of elevation is 45º. Find the height of the building.

39 DR. YOU: 018 FALL 39 LECTURE 1-4 PROPERTIES OF TRIGONOMETRIES UNIT CIRCLE SPECIAL RIGHT TRIANGLE square equilateral triangle sin(30 ) = 1 cos(30 ) = 3 tan(30 ) = sin(45 ) = cos(45 ) = tan(45 ) = 1 1 = sin(60 ) = 3 cos(60 ) = 1 tan(60 ) = 3 (, ) (, ) (, ) (, ) (, ) (, ) (, ) 0 (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, )

40 DR. YOU: 018 FALL 40 TRIGONOMETRIC FUNCTION Let θ be a real number and let P = (x, y) be the point in the plane. Let r = x + y : radius of the circle. sin θ = y r cos θ = x r tan θ = sin θ cos θ = y x csc θ = 1 sin θ = r x sec θ = 1 cos θ = r x cot θ = 1 tan θ = x y x-axis y-axis r θ (x, y) EXAMPLE 1 YOUR TURN 1 P = ( 3, 1 ) is the point on the unit circle that is corresponding to a real number θ. Find the exact value of the six trigonometric functions of θ. First, find the radius: r = ( 3 ) + ( 1 ) = = 1 = 1 P = ( 1, 6 ) is the point on the unit circle that is 5 5 corresponding to a real number θ. Find the exact value of the six trigonometric functions of θ. First, find the radius: r = x = 3, y = 1, r = 1 sin(θ) = y r = 1/ 1 = 1 sin(θ) = cos(θ) = x r = 3/ 1 = 3 cos(θ) = tan(θ) = y x = 1/ 3/ = ( 1 ) ( 3 ) = 1 3 = 3 3 csc(θ) = r y = 1 1/ = tan(θ) = csc(θ) = sec(θ) = r x = 1 3/ = (1) ( 3 ) = 3 3 sec(θ) = cot(θ) = x y = 3/ 1/ = ( 3 ) ( 1 ) = 3 cot(θ) =

41 DR. YOU: 018 FALL 41 EXAMPLE YOUR TURN P = (,) is the point on a circle that is corresponding P = (5, 1) is the point on a circle that is corresponding to a real number θ. Find the exact value of the six to a real number θ. Find the exact value of the six trigonometric functions of θ. trigonometric functions of θ. First, find the radius: r = ( ) + () = 8 = First, find the radius: r = x =, y =, r = sin(θ) = y r = = 4 = sin(θ) = cos(θ) = x r = = 4 = cos(θ) = tan(θ) = y x = = 1 tan(θ) = csc(θ) = 1 sin θ = = = csc(θ) = sec(θ) = 1 cos θ = = = sec(θ) = cot(θ) = 1 tan θ = 1 cot(θ) =

42 DR. YOU: 018 FALL 4 YOUR TURN P = ( 4,4) is the point on a circle that is corresponding to a real number θ. Find the exact value of the six trigonometric functions of θ. YOUR TURN P = ( 6, ) is the point on a circle that is corresponding to a real number θ. Find the exact value of the six trigonometric functions of θ. First, find the radius: r = First, find the radius: r = sin(θ) = sin(θ) = cos(θ) = cos(θ) = tan(θ) = tan(θ) = csc(θ) = csc(θ) = sec(θ) = sec(θ) = cot(θ) = cot(θ) =

43 DR. YOU: 018 FALL 43 NOTE: 1) A function f is called periodic if there is a positive number p such that f(θ + p) = f(p) for all θ ) Coterminal angles are angles in standard position (angles with the initial side on the positive xx -axis) that have a common terminal side. 3) If there is a smallest such number p, this smallest value is called the period of f. Period Periodic function Range sin(θ + π) = sin θ 1 sin θ 1 π cos(θ + π) = cos θ 1 cos θ 1 csc(θ + π) = csc θ (, 1] [1, ) sec(θ + π) = sec θ (, 1] [1, ) π tan(θ + π) = tan θ (, ) cot(θ + π) = cot θ (, ) EXAMPLE 3 True/ False (A) There is an angle θ such that sin θ = 3 (B) There is an angle θ such that cos θ = 3 5 (C) There is an angle θ such that tan θ = 5 (D) 10 and 150 have same values of trigonometric functions (E) 10 and 40 have same values of trigonometric functions

44 DR. YOU: 018 FALL 44 EXAMPLE 4 YOUR TURN 4 Find the exact value of the trigonometric functions of Find the exact value of the trigonometric functions of θ = 10 by using the unit circle. θ = 90 by using the unit circle. Find the terminal point of θ = 10 in the unit circle; the terminal point: ( 1, 3 ) and r = 1 Find the terminal point of θ = 90 in the unit circle; the terminal point: (, ) and r = 1 sin(10 ) = y r = 3/ 1 = 3 sin(90 ) = cos(10 ) = x r = 1/ 1 = 1 cos(90 ) = tan(10 ) = y x = 3/ 1/ = 3 tan(90 ) = csc(10 ) = 3 = 3 3 csc(90 ) = sec(10 ) = 1 = cot(10 θ) = 1 3 = 3 3 sec(90 ) = cot(90 ) =

45 DR. YOU: 018 FALL 45 EXAMPLE 5 YOUR TURN 5 Find the exact value of the trigonometric functions of Find the exact value of the trigonometric functions of θ = 135 by using the unit circle. θ = 330 by using the unit circle. Find the angle φ in 0 φ < π such that its terminal side are same to the terminal side of θ Find the angle φ in 0 φ < π such that its terminal side are same to the terminal side of θ (360 ) = 5 φ = 5 The terminal point of φ = 5 in the unit circle is ( sin( 135 ) =, ) and r = 1 The terminal point of φ = in the unit circle is the terminal point: (, ) and r = 1 sin( 330 ) = cos( 135 ) = cos( 330 ) = tan( 135 ) = = 1 tan( 330 ) = csc( 135 ) = 1 = = csc( 330 ) = sec( 135 ) = 1 = = sec( 330 ) = cot( 135 ) = = 1 cot( 330 ) =

46 DR. YOU: 018 FALL 46 EXAMPLE 6 YOUR TURN 6 Find the exact value of the trigonometric functions of Find the exact value of the trigonometric functions of θ = π 3 by using the unit circle. θ = 7π 6 by using the unit circle. Find the angle φ in 0 φ < π such that its terminal side are same to the terminal side of θ. Find the angle φ in 0 φ < π such that its terminal side are same to the terminal side of θ π 3 + (π) = 4π 3 φ = 4π 3 The terminal point of φ = 4π 3 sin ( π 3 ) = 3 in the unit circle is ( 1, 3 ) and r = 1 The terminal point of φ = in unit circle is the terminal point: (, ) and r = 1 sin ( 7π 6 ) = cos ( π 3 ) = 1 cos ( 7π 6 ) = tan ( π 3 ) = 3 1 = 3 1 = 3 tan ( 7π 6 ) = csc ( π 3 ) = 3 = 3 3 csc ( 7π 6 ) = sec ( π 3 ) = 1 = sec ( 7π 6 ) = cot ( π 3 ) = 1 3 = 3 3 cot ( 7π 6 ) =

47 DR. YOU: 018 FALL 47 EXAMPLE 7 YOUR TURN 7 Find the exact value of the trigonometric functions of Find the exact value of the trigonometric functions of θ = 405 by using the unit circle. θ = 780 by using the unit circle. Find the angle φ in 0 φ < π such that its terminal side are same to the terminal side of θ Find the angle φ in 0 φ < π such that its terminal side are same to the terminal side of θ = 45 φ = 45 The terminal point of φ = 45 in unit circle is (, ) and r = 1 The terminal point of φ = in unit circle is the terminal point: (, ) and r = 1 sin(405 ) = y r = sin(780 ) = cos(405 ) = x r = cos(780 ) = tan(405 ) = y x = / / = 1 tan(780 ) = csc(405 ) = r y = = = sec(405 ) = r x = = = cot(405 ) = x y = 1 csc(780 ) = sec(780 ) = cot(780 ) =

48 DR. YOU: 018 FALL 48 EXAMPLE 7 YOUR TURN 7 Find the exact value of the trigonometric functions of θ = 11π 3 by using the unit circle. Find the exact value of the trigonometric functions of θ = 19π 6 by using the unit circle. We know that 11π 3 = 660 Find the angle φ in 0 φ < π such that its terminal side are same to the terminal side of θ Find the angle φ in 0 φ < π such that its terminal side are same to the terminal side of θ 300 = φ = 300 The terminal point of φ = 300 in unit circle is ( 1, 3 ) and r = 1 The terminal point of φ = in unit circle is (, ) and r = 1 sin ( 11π 3 ) = y r = 3 sin ( 19π 6 ) = cos ( 11π 3 ) = x r = 1 tan ( 11π 3 ) = y x = 3 csc ( 11π 3 ) = r y = 3 = 3 sec ( 11π 3 ) = r x = cos ( 19π 6 ) = tan ( 19π 6 ) = csc ( 19π 6 ) = sec ( 19π 6 ) = cot ( 19π 6 ) = cot ( 11π 3 ) = 1 3 = 3 3

49 DR. YOU: 018 FALL 49 EXAMPLE 9 YOUR TURN 9 Find the exact value of each expression (A) cos( 135 ) (B) csc ( π 6 ) Find the exact value of each expression (A) cos ( 3π 4 ) (A) Find the angle φ in 0 φ < π such that its terminal side are same to the terminal side of θ (360 ) = 5 φ = 5 The terminal point of φ = 5 in the unit circle (B) tan( 10 ) is (, ) and r = 1. cos( 135 ) = x r = (A) Find the terminal point of π 6 = 30 ; the terminal point: ( 3, 1 ) and r = 1 (C) sec ( 17π 6 ) csc ( π 6 ) = 1 sin ( π 6 ) = 1 1/ = (D) csc ( 4π 3 )

50 DR. YOU: 018 FALL 50 NOTE: You will not be using your calculator very much in this course, but on occasion you will need to be able to find decimal approximations of the trig functions of angles (for instance, angles other than 30, 45, and 60 ). When using your calculator, you just have to set the MODE to either radians or degrees, depending on the information given in the problem. Also, your calculator has buttons for sine, cosine, and tangent, but NOT for cosecant, secant, and cotangent. So, to enter these reciprocal functions, you have to put them in as follows: csc θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = 1/tan θ. EXAMPLE 10 YOUR TURN 10 Use a calculator to find the approximate value of the expression rounded to two decimal places. csc(3 ) Use a calculator to find the approximate value of the expression rounded to two decimal places. (A) sin(8 ) 1) Check Mode in your calculator: Degree ) Plug in 1 sin(3 ) csc(3 ) = 1 sin(3 ) (B) cot(14 ) EXAMPLE 11 YOUR TURN 11 Use a calculator to find the approximate value of the expression rounded to two decimal places. sec() Use a calculator to find the approximate value of the expression rounded to two decimal places. (A) tan ( π 10 ) 1) Check Mode in your calculator: radian ) Plug in 1 cos() sec() = 1 cos().4030 (B) sec( 0.6)

51 DR. YOU: 018 FALL 51 HOMEWORK 4 NAME As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to this coversheet. 1. Find the exact values of the six trigonometric functions of the angle 480. If any are not defined, say not defined. Do not use a calculator.. Find the exact values of the six trigonometric functions of the angle 11π. If any are not defined, say not defined. Do not use a calculator Find the exact value of each expression. Do not use a calculator. (A) sin(405 ) (B) sec ( 5π 6 ) (C) tan ( π 3 ) (D) cos ( 9π 4 ) 4. Use a calculator to find the approximate value of the expression rounded to two decimal places. (A) cos(3 ) (B) csc() 5. P( 7,4) is the point on the unit circle that corresponds to a real number θ. Find the exact values of the six trigonometric functions of θ. 6. P(6, 6) is the point on the unit circle that corresponds to a real number θ. Find the exact values of the six trigonometric functions of θ. 7. P( 1, 3) is the point on the unit circle that corresponds to a real number θ. Find the exact values of the six trigonometric functions of θ.

52 DR. YOU: 018 FALL 5 LECTURE 1-5 PROPERTIES OF TRIGONOMETRIES NOTE: How to decide the quadrant of the angle θ? The positive sign of sin θ csc θ cos θ sec θ quadrant I, II I, IV (, + ) sine > 0 cosecant >0 (, ) ( +, + ) all tri functions are positve ( +, ) tan θ cot θ I, III tangent > 0 cotangent >0 cosine > 0 secant >0 EXAMPLE 1 YOUR TURN 1 Name the quadrant in which the angle θ lies Name the quadrant in which the angle θ lies cos θ > 0 ; tan θ < 0 sin θ > 0; cos θ < 0 cos θ > 0 tan θ < 0 Possible Quadrant I, IV II, IV Common quadrant IV Then the angle is in IV quadrant. EXAMPLE YOUR TURN Name the quadrant in which the angle θ lies Name the quadrant in which the angle θ lies csc θ < 0; tan θ > 0 sec θ > 0; sin θ > 0 csc θ < 0 tan θ > 0 Possible Quadrant III, IV I, III Common quadrant III Then the angle is in III quadrant.

53 DR. YOU: 018 FALL 53 EXAMPLE 3 EXAMPLE 4 Name the quadrant in which the angle θ lies tan θ > 0; sin θ < 0 Name the quadrant in which the angle θ lies csc θ < 0; cot θ < 0 EXAMPLE 5 YOUR TURN 5 Find the exact value of the four remaining trigonometric functions of θ if sin θ = 5 5 and cos θ =. 5 5 Find the exact value of the four remaining trigonometric functions of θ if sin θ = 3 5 and cos θ = ) Decide the quadrant which the terminal side of θ lies and decide the signs of x and y. 1) Decide the quadrant which the terminal side of θ lies and decide the signs of x and y. sin θ < 0 cos θ < 0 Possible Quadrant III, IV II, III Common quadrant III x =, y =, r = + ) Find the values of x, y, and r : sin θ = 5 5 = y 5 ; cos θ = r 5 = x r x = 5, y = 5, r = 5 x =, y =, r = + ) Find the values of x, y, and r : sin θ = 3 5 = y ; cos θ = 4 r 5 = x r x =, y =, r = + tan(θ) = 5 5 = 1 csc(θ) = r y = 5 5 = 5 sec(θ) = r x = 5 5 = 5 cot(θ) = 1 = tan(θ) = csc(θ) = sec(θ) = cot(θ) =

54 DR. YOU: 018 FALL 54 EXAMPLE 6 YOUR TURN 6 Find the exact value of the four remaining trigonometric functions of θ if sin θ = 5 13 and tan θ = 5 1 Find the exact value of the four remaining trigonometric functions of θ if cos θ = 3 5 and tan θ = 4 3 1) Decide the quadrant which the terminal side of θ lies and decide the signs of x and y. Possible Quadrant Common quadrant sin θ > 0 I, II II tan θ < 0 II, IV 1) Decide the quadrant which the terminal side of θ lies and decide the signs of x and y. x =, y = +, r = + x =, y =, r = + ) Find the values of x, y, and r : ) Find the values of x, y, and r : sin θ = 5 13 = y r ; tan θ = 5 1 = y x cos θ = 3 5 = x r ; tan θ = 3 4 = = y x x = 1, y = +5, r = +13 x =, y =, r = + cos(θ) = x r = 1 13 csc(θ) = 13 5 sec(θ) = 13 1 cot(θ) = 1 5 cos(θ) = csc(θ) = sec(θ) = cot(θ) =

55 DR. YOU: 018 FALL 55 EXAMPLE 7 YOUR TURN 7 Find the exact value of the four remaining trigonometric Find the exact value of the four remaining trigonometric functions of θ; functions of θ; θ is an acute angle and cos θ = 4 5 θ is an acute angle and cos θ = 1 3 1) Decide the quadrant: Acute angle: it is in I quadrant. 1) Decide the quadrant: ) Decide the signs of x, y. x = +, y = +, r = + ) Decide the signs of x, y. x =, y =, r = + 3) Find the values of x, y, and r : 3) Find the values of x, y, and r : Since cos θ = 4 5 = x r x = + 4, y = +, r = +5 Since x + y = r y = 5 16 = 9 y = 3 because of y > 0 by ) x = 4, y = 3, r = 5 sin(θ) = y r = 3 5 sin(θ) = cos(θ) = x r = 4 5 cos(θ) = tan(θ) = y x = 3 4 tan(θ) = csc(θ) = csc(θ) = r y = 5 3 sec(θ) = sec(θ) = r x = 5 4 cot(θ) = cot(θ) = x y = 4 3

56 DR. YOU: 018 FALL 56 EXAMPLE 8 YOUR TURN 8 Find the exact value of the four remaining trigonometric functions of θ; θ is in II quadrant and sin θ = 7 5 Find the exact value of the four remaining trigonometric functions of θ; θ is in III quadrant and tan θ = ) Decide the quadrant: II quadrant 1) Decide the quadrant: ) Decide the signs of x, y. x =, y = +, r = + ) Decide the signs of x, y. x =, y =, r = + 3) Find the values of x, y, and r : 3) Find the values of x, y, and r : sin θ = 7 5 = y r x =, y = + 7, r = +5 Since x + 7 = 5, x = 576 x = 576 = 4 because of x < 0 by ) x = 4, y = 7, r = 5 sin(θ) = sin(θ) = y r = 7 5 cos(θ) = x r = 4 5 tan(θ) = y x = 7 4 csc(θ) = r y = 5 7 sec(θ) = r x = 5 4 cot(θ) = x y = 4 7 cos(θ) = tan(θ) = csc(θ) = sec(θ) = cot(θ) =

57 DR. YOU: 018 FALL 57 EXAMPLE 9 YOUR TURN 9 Find the exact value of the four remaining trigonometric functions of θ; 3π < θ < π and tan θ = 3 4 Find the exact value of the four remaining trigonometric functions of θ; π < θ < 0 and sin θ = ) Decide the quadrant: 1) Decide the quadrant: IV quadrant ) Decide the signs of x, y. x = +, y =, r = + ) Decide the signs of x, y. x =, y =, r = + 3) Find the values of x, y, and r : 3) Find the values of x, y, and r : tan θ = 3 4 = y x x = +4, y = 3, r = + Since = r r = 5 r = 5 x = 3, y = 4, r = 5 sin(θ) = sin(θ) = y r = 4 5 cos(θ) = cos(θ) = x r = 3 5 tan(θ) = tan(θ) = y x = 4 3 csc(θ) = csc(θ) = r y = 5 4 sec(θ) = sec(θ) = r x = 5 3 cot(θ) = cot(θ) = x y = 3 4

58 DR. YOU: 018 FALL 58 YOUR TURN Find the exact value of the four remaining trigonometric functions of θ; cot θ > 0 and csc θ = 13 5 YOUR TURN Find the exact value of the four remaining trigonometric functions of θ; 3π < θ < π and sec θ = 3

59 DR. YOU: 018 FALL 59 HOMEWORK 5 NAME As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to this coversheet. 1. Let θ be an angle in standard position. Name the quadrant in which θ lies (A) cos θ < 0, tan θ < 0 (B) csc θ > 0, sec θ < 0. Find the exact value of each of the remaining trigonometric function of θ (A) sin θ = 5, θ is an acute angle (B) sin θ = 3, 3 π < θ < π 13 5 (C) cos θ = 1, π < θ < π (D) tan θ = 5, sin θ > (E) cot θ = 4 3, cos θ < 0 (F) sec θ = 5, tan θ > 0

60 DR. YOU: 018 FALL 60 REVIEW PROBLEMS FOR EXAM 1 Click the blue words to see the video 1. Convert angle from radian to degree: Problem 1~. Convert angle form degrees to radian: Problem 3 3. Find exact value of trigonometric functions by using Unit circle : Problem 4 4. Find the value of trigonometric function by using a calculator: Problem 5 5. Evaluate the trigonometric function given a point in the terminal side: Problem Evaluate the trigonometric function given one trig and angle information: Problem 8,9 1) Case 1 ) Case 7. Find the arc length and sector area: Problem Application in right triangle trigonometry: Problem 11, 1 9. Find the angular speed and linear speed: Problem 13, Application (hard one) : Problem Convert 13π 1 in radians to degrees.. Convert 5 in radians to degrees. 3. Convert 35 to exact radian measure. 4. Find exact value of the following trigonometry expression without using your calculator. (A) sin ( 9π 6 ) (B) cos(510 ) (C) sec( 330 ) (D) cot (5π 3 ) 5. Use your calculator to find the following value (Round to the nearest thousandth). (A) sin(37 ) (B) cot() 6. Find the values of all six trigonometric functions of the angle, θ, which is in standard position with the terminal side passing through the point P( 8, 15).

61 DR. YOU: 018 FALL Find the values of all six trigonometric functions of the angle, θ, which is in standard position with the terminal side passing through the point P( 3, 3). 8. Given tan θ = 4 7 and cos θ < 0, find the exact values of the other five trigonometric functions. 9. Given sin θ = 5 and π < θ < π, find the exact values of the other five trigonometric functions Find the length of arc in inches and the area of sector in square inches subtended by a central angle of 60 on a circle of radius 10 inches. 11. A ladder 0 feet long is leaning against a building. The angle formed by the ladder to the ground is 55. How far from the base of the building is the base of the ladder? 1. A tree casts a shadow of 45 meters when the angle of elevation of the sun is 5. Find the height of the tree to the nearest hundredth of a meter. 13. A wheel with the radius 10 inches is rotating at 300 revolutions per second. Find the angular velocity in radians per second and find the linear speed in feet per minute. 14. Cathy is riding a bike with 14-inch diameter wheels which are rotating 3 revolutions per second. What is her linear speed in feet per minute? (Round to the nearest hundredth of a foot per minute and use the π key on your calculator to perform the computation.) 15. At a certain distance, the angle of elevation to the top of a building is 60º. From 40 feet further back, the angle of elevation is 45º. Find the height of the building

62 DR. YOU: 018 FALL 6 SOLUTIONS π 1 = 13π π = = = π = 900 π π 35 = = 35 π 180 = π (A) 9π = 4π + 5π : sin (9π ) = sin (5π) = 1 6 since the terminal point in the unit circle is ( 3, 1 ) 5. Check mode (B) 510 = : cos(510 ) = cos(150 ) = 3 is ( 3, 1 ). since the terminal point in the unit circle (C) 330 = : sec( 330 ) = sec(30 ) = 1 = = 3 since the terminal cos(30 ) 3 point in the unit circle is ( 3, 1 ). (D) cot ( 5π 3 ) = x y = 1 3 = 1 () (A) Mode(degree): sin(37 ) = 1 = 3 3 () 3 3 (B) Mode(radian): cot() = 1 tan() r = ( 8) + ( 15) = 89 = 17 sin θ = y r = ; cos θ = x r = 8 17 ; tan θ = y x = ; csc θ = ; sec θ = ; cot θ = r = ( 3) + (3) = 18 = 3 sin θ = y r = 3 3 = 1 = ; cos θ = x r = 3 3 = 1 = ; tan θ = y x = 3 3 = 1; csc θ = 3 3 = ; sec θ = 3 3 = ; cot θ = 3 3 = ) tan θ = 4 7 sin θ < 0 cos θ = and cos θ < 0 sin θ x < 0, y < 0 which is in III quadrant ) tan θ = 4 7 = 4 7 = y x x = 7, y = 5, r = ( 7) + ( 4) = 65 = 5 sin θ = y r = 4 5 ; cos θ = x r = 7 5 ; tan θ = y x = ; csc θ = ; sec θ = ; cot θ = 7 4

63 DR. YOU: 018 FALL ) π < θ < π is in II quadrant. x < 0, y > 0 ) sin θ = 5 13 = y r y = 5, r = 13 3) x + y = r x + (5) = (13) x = 144 x = ±1, x < 0 by 1) x = 1 Then x = 1, y = 5, r = 13 sin θ = y r = 5 13 ; cos θ = x r = 1 13 ; tan θ = y x = ) θ = 60 = Let h= height π 180 = π 3 ) The length of arc is rθ = 10 π 3 = 10π 3 in 3) The area of the sector = 1 r θ = 1 10 π 3 = 50π 3 in ; csc θ = ; sec θ = ; cot θ = 1 5 sin(55 ) = h 0 h = 0 sin(55 ) ft 1. Let h = height 4 tan(4 ) = h 6 h = 6 tan(4 ) m Angular speed = π the number of rotation per unit time = π 300 = 600π rad/sec Linear speed = r angular speed = π = 6,000π in/sec = 6000π 1 60 = 30,000 ft/min Radius = 7 in 15. Angular speed = π the numer of rotation per unit time = π 3 = 6π rad/sec Linear speed = r angular speed = 7 6π = 4π in/sec 4π in 1 sec 1 ft 60 sec = 10π ft/min ft/min 1 in 1 min tan(60 ) = h y ; tan(45 ) = h y + 40 h = y tan(60 ) ; (y + 40) tan(45 ) = h (y + 40) tan(45 ) = y tan(60 ) y tan(45 ) + 40 tan(45 ) = y tan(60 ) y tan(45 ) y tan(60 ) = 40 tan(45 ) y = 40 tan(45 ) tan(45 ) tan(60 ) ; h = y tan(60 ) = 40 tan(45 ) tan(60 ) = ft tan(45 ) tan(60 ) 1

64 DR. YOU: 018 FALL 64 CHAPTER. PROPERTIES OF TRIG LECTURE -1 SINE AND CONSINE FUNCTIONS THE GRAPH OF TRIGONOMETRIC FUNCTIONS Let n be integers. y = sin x y = cos x Domain (, ) (, ) Range [ 1,1] [ 1,1] x intercept πn πn + π y intercept 0 1 Symmetry origin y-axis THE AMPLITUDE AND PERIOD: y = A sin(bx + c) + d ; y = A cos(bx + c) + d 1) The amplitude of a sinusoidal function is onehalf of the positive difference between the maximum and minimum values of a function Amplitude: A ) A periodic function is an oscillating (wave-like) function which repeats a pattern of y-values at regular intervals. base line period : π b phase shift : c b vertical shift : d y = A sin(bx + c) + d amplitude: A Period: π b 3) A phase shift is the horizontal shift Phase shift: c b 4) Midline (vertical shift): y = d

65 DR. YOU: 018 FALL 65 EXAMPLE 1 How to draw the graph of sine; How to draw the graph of cosine : Click to see the video Draw the graph of y = sin(x) x: angle 0 π 6 = 30 π = 90 5π 6 = 150 π = 180 7π 6 = 10 3π = 70 11π 6 = 330 π = 360 y = sin(x) EXAMPLE Draw the graph of y = cos(x) x: angle y = cos(x) 0 π 3 = 60 π = 90 π 3 = 10 π = 180 4π 3 = 40 3π = 70 5π 3 = 300 π = 360

66 DR. YOU: 018 FALL 66 EXAMPLE 3 YOUR TURN 3 Determine the amplitude, period, phase shift, domain, and range of each function without graphing. y = 5 sin ( 3x 5 π 3 ) Determine the amplitude, period, phase shift, domain, and range of each function without graphing. (A) y = 5 sin(3x) We find A, b, c and d. 5 sin ( 3x 5 π 3 ) = 5 sin (3 5 x + ( π 3 )) A = 5, b = 3 5, c = π 3 Amplitude: Period: Phase shift: Amplitude: A = 5 = 5 Vertical shift: Period: Phase shift: π b = π ( 3 = π ( 5 5 ) 3 ) = 10π 3 Domain: Range: (B) y = 3 sin ( x π) c b = π = π 3 (5 3 ) = 5π 9 Domain: (, ) Range: [ 5,5] Vertical shift: None Amplitude: Period: Phase shift: Vertical shift: Domain: Range:

67 DR. YOU: 018 FALL 67 YOUR TURN Determine the amplitude, period, phase shift, domain, and range of each function without graphing (A) y = 7 sin(πx + 4) YOUR TURN Determine the amplitude, period, phase shift, domain, and range of each function without graphing (A) y = 4 cos ( x ) Amplitude: Period: Phase shift: Amplitude: Period: Phase shift: Vertical shift: Domain: Range: Vertical shift: Domain: Range: (B) y = 5 sin ( x 3 + 6) (B) y = 4 cos(πx 4) Amplitude: Period: Phase shift: Amplitude: Period: Phase shift: Vertical shift: Domain: Range: Vertical shift: Domain: Range:

68 DR. YOU: 018 FALL 68 EXAMPLE 4 Draw the graph of y = 3 sin(x π). 1) Find the amplitude, Period, and Phase shift. Amplitude: 3 = 3 Period: π b = π () = π Phase shift: c b = π = π ) To find the five key points, find t = period 4 = π 4 Five x-values: 0, π, π, 3π, 4π Find the corresponding y-values: it starts 0. 0, 3, 0, 3, 0 (0, 0) ( π 4, 3) (π 4, 0) (3π 4, 3) ( 4π 4, 0) Five points in y = 3 sin(x π): Add (phase shift, Vertical shift) ( π, 0) (3π 4, 3) ( 4π 4, 0) (5π 4, 3) ( 6π 4, 0) 3) Pick those five points and draw the graph 4 y = 3 sin( x π) π π 3π 4

69 DR. YOU: 018 FALL 69 EXAMPLE 5 Draw the graph of y = 5 sin ( x π 4 ).

70 DR. YOU: 018 FALL 70 EXAMPLE 6 Draw the graph of y = cos ( x + π) ) Find the amplitude, Period, and Phase shift. Amplitude: = Period: π = π = 4π b (1/) Phase shift: c b = π 1 Vertical shift: d = +1 = π ) To find the five key points of y = cos ( x Five x-values: 0, π, π, 3π, 4π Find the corresponding y-values: it starts + π) + 1, find t = period 4, 0,, 0, = 4π 4 = π (0, ) (π, 0) (π, ) (3π, 0) (4π, ) Five points in y = cos ( x + π) + 1 Add (phase shift, Vertical shift) = ( π, 1) ( π, 3) ( π, 1) (0, 1) (π, 1) (π, 3) y = cos ( ) + 1 x + π 4 π π π π 3π 4π

71 DR. YOU: 018 FALL 71 EXAMPLE 7 Draw the graph of y = 3 cos(x + π).

72 DR. YOU: 018 FALL 7 EXAMPLE 8 YOUR TURN 8 Write the equation of a sine function y = A sin(bx + c), b > 0 that has the given characteristics Amplitude: 3 Period: π Phase shift: 1 Write the equation of a sine function y = A sin(bx + c), b > 0 that has the given characteristics Amplitude: 5 Period: 4π Phase shift: π y = A sin(bx + c) 1) A = 3 ) Find b > 0 : π b = π π = bπ b = 3) Find b by using phase shift c b = 1 c = 1 c = c = 1 When A > 0: A = 3 When A > 0: Therefore, y = 3 sin(x 1) When A < 0: A = 3 When A < 0: Therefore, y = 3 sin(x 1)

73 DR. YOU: 018 FALL 73 YOUR TURN Write the equation of a sine function y = A sin(bx + c), b > 0 that has the given characteristics Amplitude: 3 Period: π Phase shift: YOUR TURN Write the equation of a sine function y = A sin(bx + c), b > 0 that has the given characteristics Amplitude: 7 Period: 3π Phase shift: 1 3

74 DR. YOU: 018 FALL 74 EXAMPLE 9 YOUR TURN 9 Find an equation y = A sin(bx + c) where A > 0 and b > Find an equation y = A sin(bx + c) where A > 0 and 0 for the graph below: b > 0 for the graph below: (A) 4 π π π π 4 First, find the first negative x-intercept point 4 period π π π π π Phase shift π Amplitude 3 (B) Then 1) A = 3 ) π b = π b = 3) c b = π 4 c = π 4 c = π Therefore, y = 3 sin (x π )

75 DR. YOU: 018 FALL 75 EXAMPLE 10 YOUR TURN 10 A circle with radius 3 ft. is mounted with its center 4-ft off The London Eye is a huge wheel with a diameter of 135 the ground. The point closest to the ground is labeled P. meters (443 feet). It completes one rotation every 30 Find the function that gives the height is terms of the minutes. Riders board from a plate meters above the angle of rotation. ground. Express a rider s height above ground as a function of time in minutes. 3 ft 3 ft P 1 ft Then, we can get the height information in terms of the angle of rotation. The angle The height 0 1 ft 90 4 ft ft 70 4 ft ft Period: π Midline: 4 ft Amplitude: highest point midpoint = 3 ft Phase shift: 90 = π So, y = A sin(bx + c) A = 4 b = 1 since the period is π = π b b = 1 c = π since c b = π c = π c = π d = 4 because of the midline y = 4 sin (x π ) + 4

76 DR. YOU: 018 FALL 76 HOMEWORK 6 NAME As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to this coversheet. 1. Find the amplitude, the period, and the phase shift and sketch the graph of the equation (A) y = 3 sin (x + π 6 ) (B) y = 7 cos (x π 4 ) (C) y = sin (πx π ) (D) y = 5 cos (x π). Write the equation of a sine function that has the given characteristics. (A) Amplitude: and period: 4π (B) Amplitude: and period: π; phase shift: 1 3. The graph of an equation is shown in the figure. Write the equation in the form y = A sin(bx + c) for A > 0 and b > 0. (A) (B)

77 DR. YOU: 018 FALL 77 LECTURE - THE OTHER TRIGONOMETRIC FUNCTIONS GRAPH OF TANGENT: Let n be arbitrary integer. The graph of y = tan x is f( x) = tan( x) 4 Vertical asymptotes are x = π + nπ, n Z TANGENT π π π π 4 Domain: all real numbers except π + nπ, n Z Range: (, ) Period: π Let y = A tan(bx + c) + d Period Phase shift Vertical shift Vertical Asymptote π b c b d x = c b π(n + 1) +, n Z b The graph of y = cot(x) is CONTANGENT 4 f (x) = cot (x) π π π π Vertical asymptotes are x = nπ, n Z Domain: all real numbers except nπ, n Z Range: (, ) Period: π 4 Let y = A cot(bx + c) + d Period Phase shift Vertical shift Vertical Asymptote π b c b d x = c b + πn b, n Z

78 DR. YOU: 018 FALL 78 How to draw the graph of tangent : Click to see the video EXAMPLE 1: We know the graph of y = tan x; 4 y = tan( x) π π π π period π 4 Period: the graph of y = tan(x) (period is π ) is 4 y = tan( x) π π π π 4 Period: the graph of y = tan ( x π ) (period is π = ) is 1/ 4 y = tan( x ) π π π π 4

79 DR. YOU: 018 FALL 79 EXAMPLE : The graph of y = tan ( x ) is 4 y = tan( x ) π π π π 4 Phase shift: the graph of y = tan ( x π ) is 4 phase shift π π π π π 4 Reflect about x-axis: the graph of y = tan ( x ) is 4 ( ) y = tan x π π π π 4

80 DR. YOU: 018 FALL 80 EXAMPLE 3 YOUR TURN 3 Find the period, range, domain, and vertical asymptotes of Find the period, domain, range, vertical asymptotes of y = tan ( x 3 ) (A) y = 3 tan(x) We know that y = tan ( x 3 ) = tan (1 3 x) 1) Find the period π b = π 1/3 = 3π ) Find the range (, ) 3) Find the vertical asymptote A vertical asymptote is x = period + phase shift = 3π The set of all vertical asymptote is (B) y = 5 tan ( x 4 ) x = 3π + (period) n, n: any integers x = 3π + (3π)n, n: any integers 4) Find the domain The domain is all real numbers except 3π + (3π)n, n: any integers

81 DR. YOU: 018 FALL 81 EXAMPLE 4 The graph of y = cot(x) is 4 period π π π π π 4 Period: the graph of y = cot(x) is 4 period π π π π π 4 Period: the graph of y = cot ( x ) is 4 π π π π period π 4

82 DR. YOU: 018 FALL 8 EXAMPLE 5 YOUR TURN 5 Find the period, domain, range, vertical asymptotes of Find the period, domain, range, vertical asymptotes of y = 5 cot(3x) (A) y = 3 cot(x) 1) Find the period π b = π 3 = π 3 ) Find the range (, ) 3) Find the vertical asymptote A vertical asymptote is x = phase shift = 0 The set of all vertical asymptote is x = 0 + (period) n, n: any integers (B) y = 6 cot ( x 3 ) x = ( π 3 ) n, n: any integers 4) Find the domain The domain is all real numbers except ( π ) n, n: any integers 3

83 DR. YOU: 018 FALL 83 THE OTHER TRIGONOMETRIC FUNCTIONS The graph of y = csc(x) is COSECANT 4 f (x) = csc (x) π π π π Vertical asymptotes are x = nπ, n Z Domain: all real numbers except nπ, n Z Range: (, 1] [1, ) Period: π 4 Let y = A csc(bx + c) + d whose range is (, A ] [ A, ) Period Phase shift Vertical shift Vertical Asymptote π b c b d x = c b + πn b, n Z The graph of y = sec(x) is SECANT 4 f (x) = sec (x) π π π π Vertical asymptotes are x = π + nπ, n Z Domain: all real numbers except π + nπ, n Z Range: (, 1] [1, ) Period: π 4 Let y = A csc(bx + c) + d whose range is (, A ] [ A, ). Period Phase shift Vertical shift Vertical Asymptote π b c b d x = c b π(n 1) +, n Z b

84 DR. YOU: 018 FALL 84 EXAMPLE 6 The graph of y = csc(x) is in the below figure such that the period is π and range is (, 1] [1, ). It has vertical asymptotes at x = 0, ±π, ±π, ±3π,. 4 y =sin(x) π π π π 4 Period: the graph of y = csc(x) is in the below figure such that the period is π = π. It has vertical asymptotes at x = 0, ± π, ±π, ± 3π,. 4 π π π π 4 Range: the graph of y = 3 csc(x) is in the below figure such that range is (, 3] [3, ) 4 π π π π 4 6

85 DR. YOU: 018 FALL 85 EXAMPLE 7 YOUR TURN 7 Find the period, phase shift, range, and vertical Find the period, phase shift, range, and vertical asymptotes of asymptotes of y = 3 csc(3x 1) (A) y = csc(x) 1) Find A, b, and c A = 3, b = 3, c = 1 ) Period: π b = π 3 3) Phase shift: c b = 1 3 = 1 3 4) Find the range: It is related to A = 3 (, 3] [3, ) (B) y = 5 csc ( x ) 5) Find the vertical asymptotes A vertical asymptote is x = phase shift = 1 3 The set of all vertical asymptote is x = (period ) n, n: any integers x = (π 3 ) n, n: any integers 6) Find the domain: The domain is all real numbers except (π ) n, n: any integers 3

86 DR. YOU: 018 FALL 86 EXAMPLE 8 The graph of y = sec(x) is in the below figure such that the period is π and range is (, 1] [1, ). It has vertical asymptotes at x = ± π, ± 3π, ± 5π,. 4 π π π π 4 Period: the graph of y = sec ( x π ) is in the below figure such that the period is = 4π. 1/ It has vertical asymptotes at x = 0, ±π, ±3π, ±5π,. 4 π π π π 4 Range: the graph of y = 3 sec(x) is in the below figure such that range is (, 3] [3, ) 4 π π π π 4

87 DR. YOU: 018 FALL 87 EXAMPLE 9 YOUR TURN 9 Find the period, phase shift, range of Find the period, domain, range, asymptote of y = 4 sec(x π) (A) y = 4 sec(x) 1) Find A, b, and c A = 4, b =, c = π ) Period: π b = π = π 3) Phase shift: c b = π = π 4) Find the range: It is related to A = 4 (, 4] [4, ) (B) y = sec ( x 4 ) 5) Find the vertical asymptotes A vertical asymptote is x = phase shift = π The set of all vertical asymptote is x = ( period ) (n 1), n: any integers 4 x = π (n 1), 4 n: any integers 6) Find the domain: The domain is all real numbers except π (n 1), n: any integers 4

88 DR. YOU: 018 FALL 88 EXAMPLE 10 YOUR TURN 10 Find all x in π x π such that sin x = 1. Find all x in π x π such that cos x = 1. First, solve sin x = 1 on 0 x < π x = π Since sin x has period π, there are two cycles. The set of solutions are x = π, x = π π = 3π EXAMPLE 11 YOUR TURN 11 Find all x in π x π such that sec x = 1. Find all x in π x π such that csc x = 1. First, solve cos x = 1 on 0 x < π since sec x = 1 cos x x = 0 Since cos x has period π, The set of solutions are x = 0 + πn, n any integer Find all solutions in π x π x = π, 0, π

89 DR. YOU: 018 FALL 89 HOMEWORK 7 NAME As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to this coversheet. 1. Find the period and all vertical asymptotes of the graph of the equation. (A) y = 4 tan(x) (B) y = 5 cot ( x ). For what number x, π x π, does csc(x) = 1? 3. For what number x, π x π, does the function f(x) = cot(x) have vertical asymptotes? x-intercepts? 4. Find the period, the phase shift, and the range. (A) y = tan (x π ) (B) y = 5 cos 4 (x + π ) 3 4 (C) y = 3 sin (πx π ) (D) y = csc (x + π )

90 DR. YOU: 018 FALL 90 LECTURE -3 INVERSE TRIGONOMETRIC FUNCTIONS INVERSE The function Inverse function Domain of inverse Range of inverse y = sin x y = sin 1 x or y = arcsin x [ 1,1] [ π, π ] y = cos x y = cos 1 x or y = arccos x [ 1,1] [0, π] y = tan x y = tan 1 x or y = arctan x (, ) ( π, π ) y = csc x y = csc 1 x or y = arccsc x (, 1] [1, ) [ π, π ], y 0 y = sec x y = sec 1 x or y = arcsec x (, 1] [1, ) [0, π], y π y = cot x y = cot 1 x or y = arccot x (, ) (0, π) NOTE: csc 1 x = sin 1 ( 1 x ), sec 1 x = cos 1 ( 1 x ), cot 1 x = tan 1 ( 1 x )

91 DR. YOU: 018 FALL 91 Evaluating the inverse of trigonometric functions without a calculator : Click to see the video Evaluate the composition of trigonometric functions: 1) Case 1 : inverse trig (trig) ) Case : trig (inverse trig) EXAMPLE 1 YOUR TURN 1 Find the exact value in radian of (A) sin 1 ( 1 ) (B) sin 1 ( 3 ) Find the exact value in radian of (A) sin 1 (1) (A) Since θ = sin 1 ( 1 ) sin θ = 1 and π θ π Then θ is in I quadrant. Find θ in I quadrant such that sin θ = 1 θ = sin 1 ( 1 ) = π 6 (B) Since θ = sin 1 ( 3 ) (B) sin 1 ( ) sin θ = 3 and π θ π. Then, θ is in IV quadrant. Find θ in π θ 0 and sin θ = 3 sin 1 ( 3 ) = π 3 (Warning: it is not 5π since π θ 0 ) 3

92 DR. YOU: 018 FALL 9 EXAMPLE YOUR TURN Find the exact value in radian of Find the exact value in radian of (A) tan 1 (1) (B) tan 1 ( 3) (A) tan 1 ( 1 ) 3 (A) Since θ = tan 1 (1) tan θ = 1 and π θ π, Then, θ is in I quadrant. Find θ in I quadrant such that tan θ = 1. tan 1 (1) = π 4 (B) tan 1 ( 1) (B) Since θ = tan 1 ( 3) tan θ = 3 and π θ π, Then, θ is in IV quadrant. Find θ in π θ 0 such that sin θ = 3 tan 1 ( 3) = π 3 (C) tan 1 ( 3) (Warning: it is not 5π since π θ 0 ) 3

93 DR. YOU: 018 FALL 93 EXAMPLE 3 YOUR TURN 3 Find the exact value in radian of (A) cos 1 ( ) (B) cos 1 ( 3 ) Find the exact value in radian of (A) cos 1 ( 3 ) (A) Since θ = cos 1 ( ) cos θ = Then, θ is in I quadrant and 0 θ π, Find θ in I quadrant such that cos θ =. cos 1 ( ) = π 4 (B) cos 1 ( 1 ) (B) Since θ = cos 1 ( 3 ) cos θ = 3 Then, θ is in II quadrant and 0 θ π, (C) cos 1 (1) Find θ in II quadrant such that cos θ = 3. cos 1 ( 3 ) = 5π 6

94 DR. YOU: 018 FALL 94 YOUR TURN Find the exact value in radian of (A) sin 1 ( 1 ) (B) cos 1 ( 1 ) (C) cos 1 ( 1) (D) tan 1 ( 3) (E) sin 1 ( 3 ) (F) tan 1 (0)

95 DR. YOU: 018 FALL 95 EXAMPLE 4 YOUR TURN 4 Use a calculator to find exact value of the following in degree. Round four decimal places. (A) sin 1 ( 4 9 ) (B) cos 1 ( 1 13 ) (C) sin 1 (1.) (D) tan 1 (4) Use a calculator to find exact value of the following in radian. Round four decimal places. (A) sin 1 ( 5 9 ) (B) cos 1 ( 0.6) (A) sin 1 ( 4 9 ) (B) cos 1 ( 1 13 ) (C) sin 1 (1.) is undefined: error since there are no angles θ such that sin θ = 1.. (D) tan 1 (4) = (C) cos 1 () (D) tan 1 (6) EXAMPLE 5 YOUR TURN 5 Use a calculator to find exact value of the following in degree. Round four decimal places. Use a calculator to find exact value of the following in degree. Round four decimal places. (A) csc 1 ( 3 ) (B) sec 1 ( 5 ) 3 (A) csc 1 (1.) (B) cot 1 (3) (A) We find angle θ = csc 1 ( 3 ) such that csc θ = 1 sin θ = 3 sin θ = 3 θ = sin 1 ( 3 ) = csc 1 ( 3 ) csc 1 ( 3 ) = sin 1 ( 1 3/ ) = (B) We find angle θ = sec 1 ( 5 ) such that 3 sec θ = 1 cos θ = 5 3 cos θ = 3 5 θ = cos 1 ( 3 5 ) sec 1 ( 5 3 ) = cos 1 ( 1 5/3 ) =

96 DR. YOU: 018 FALL 96 EXAMPLE 6 YOUR TURN 6 Find the angle θ in degree in the figure Find the angle θ in degree in the figure 1 6 θ 9 θ 7 Hypotenuse: 1 Adjacent: 9 cos θ = 9 1 θ = cos 1 ( 9 1 ) EXAMPLE 7 YOUR TURN 7 Find the angle θ in degree in the figure Find the angle θ in degree in the figure 7 θ θ Hypotenuse: 10 Opposite side: 6 sin θ = 6 10 θ = sin 1 ( 6 10 ) 36.87

97 DR. YOU: 018 FALL 97 THE RELATION BETWEEN INVERSE FUNCTIONS Functions relation Sine function Cosine function sin 1 (sin θ) = θ, π θ π sin(sin 1 A) = A, 1 A 1 cos 1 (cos θ) = θ, 0 θ π cos(cos 1 A) = A, 1 A 1 Tangent function tan 1 (tan θ) = θ, π < θ < π tan(tan 1 A) = A, < A < I quadrant II quadrant III quadrant IV quadrant 0 θ < π sin 1 (sin θ) θ π θ π θ 0 θ < π tan 1 (tan θ) θ θ π θ π π θ < π cos 1 (cos θ) θ θ EXAMPLE 8 YOUR TURN 8 Find the exact value in radian of Find the exact value in radian of (A) cos 1 (cos ( π 4 )) (B) cos 1 (cos ( 9π 8 )) (A) cos 1 (cos ( 5π 4 )) First, find a coterminal angle in 180 < θ < 180 (A) cos 1 (cos ( π )) 4 = cos 1 (cos( 45 )) = 45 = 45 = π 4 (B) cos 1 (cos ( 3π 5 )) (B) cos 1 (cos ( 9π )) 8 = cos 1 (cos(0.5 )) = cos 1 (cos( )) = = = 7π 8

98 DR. YOU: 018 FALL 98 EXAMPLE 9 YOUR TURN 9 Find the exact value in radian of Find the exact value in radian of (A) tan 1 (tan ( 5π 4 )) (A) tan 1 (tan ( π 3 )) = tan 1 (tan(5 )) = since 180 < 5 < 70 = 45 = π 4 (B) tan 1 (tan ( π 3 )) = tan 1 (tan( 60 )) (B) tan 1 (tan ( 11π 6 )) = 60 since 90 < 60 < 0 = π 3 (C) tan 1 (tan(110 )) = since 90 < 110 < 180 = 70 (C) tan 1 (tan ( π 5 ))

99 DR. YOU: 018 FALL 99 EXAMPLE 10 YOUR TURN 10 Find the exact value in radian of Find the exact value in radian of (A) sin 1 (sin ( π 3 )) (A) sin 1 (sin ( 4π 3 )) = sin 1 (sin(60 )) = 60 since 90 < 60 < 90 = π 3 (B) sin 1 (sin ( 6π 5 )) = sin 1 (sin(16 )) = since 16 is in III quadrant. (B) sin 1 (sin ( π 6 )) = 36 = π 5 (C) sin 1 (sin ( 5π 8 )) = sin 1 (sin(11.5 )) = since 11.5 is in II quadrant. = 67.5 (C) sin 1 (sin ( 3π 5 )) = 3π 8

100 DR. YOU: 018 FALL 100 EXAMPLE 11 YOUR TURN 11 Find the exact value without a calculator of Find the exact value without a calculator of (A) cos (tan 1 ( 5 1 )) (B) tan (cos 1 ( 5 13 )) (A) cos (tan 1 ( 5 1 )) (A) Let θ = tan 1 ( 5 1 ), π θ π. Since tan θ = + 5 and π θ π, the angle 1 lies in I quadrant. x y r = x + y cos (tan 1 ( 5 1 )) = cos θ = 1 13 (B) tan (cos 1 ( 3 5 )) (B) Let θ = cos 1 ( 5 ) where 0 θ π 13 Since cos θ = 5 and 0 θ π, 1 x = 5, r = 13 in II quadrant Since ( 5) + y = 13, and y > 0, y = 144 y = 1 (C) cot (sin 1 ( 7 5 )) x y r = x + y tan (cos 1 ( 5 1 )) = tan θ = 13 5

101 DR. YOU: 018 FALL 101 EXAMPLE 1 Find the exact value without a calculator of (A) sin 1 (sin ( 3π 8 )) (B) cos 1 (cos ( 13π 1 )) (C) cos 1 (sin ( 7π 10 )) (D) tan 1 (cos ( 5π 8 )) (E) cos (sin 1 ( 8 17 )) (F) sin (tan 1 ( 5 4 ))

102 DR. YOU: 018 FALL 10 HOMEWORK 8 NAME As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to this coversheet. 1. Find the exact value of the expression whenever it is defined (A) sin 1 ( 3 ) (B) cos 1 ( 1 ) (C) tan 1 ( 1 3 ) (D) cos 1 ( ). Find the exact value of the expression whenever it is defined (A) sin 1 (sin ( 6π 5 )) (B) cos 1 (cos ( 11π 8 )) (C) tan 1 (tan ( 7π 4 )) (D) sin 1 (cos ( π 3 )) 3. Find the exact value of the expression whenever it is defined (A) cos (sin 1 ( 4 5 )) (B) cot (sin 1 ( 3 )) (C) tan (cos 1 ( 1 13 )) (D) csc (tan 1 ( 3 4 )) 4. Suppose a 13-foot ladder is leaning against a building, reaching to the bottom of a second-floor window 1 feet above the ground. What angle, in radians, does the ladder make with the building?

103 DR. YOU: 018 FALL 103 LECTURE -4 IDENTITIES IDENTITY OF TRIGONOMETRY: csc θ = 1 sin θ sec θ = 1 cos θ tan θ = sin θ cos θ cot θ = 1 tan θ = cos θ sin θ sin θ + cos θ = 1 tan θ + 1 = sec θ cot θ + 1 = csc θ Tips to verifying trig identities : Click to see video EXAMPLE 1 YOUR TURN 1 Find the exact value of the expression without calculator. tan(35 ) sin(35 ) cos(35 ) Find the exact value of each expression without calculator. (A) cot(x) cos(x) sin(x) We know that tan(35 ) = sin(35 ) cos(35 ) Therefore, tan(35 ) sin(35 ) cos(35 ) (B) sin(θ) csc(θ) = sin(35 ) cos(35 ) sin(35 ) cos(35 ) = 0 (C) tan(71 ) cot(71 ) (D) sec(θ) cos(θ)

104 DR. YOU: 018 FALL 104 EXAMPLE YOUR TURN Find the exact value of the expression without calculator. cos 1 (θ) + csc (θ) Find the exact value of each expression without calculator. (A) sin (θ) + cos (θ) We know that csc(θ) = 1 sin(θ) ; 1 csc (θ) = sin (θ) Therefore, cos 1 (θ) + csc (θ) (B) sin (50 ) + cos (50 ) = cos (θ) + sin (θ) = 1 (C) sec (θ) tan (θ) (D) cot (33 ) csc (33 )

105 DR. YOU: 018 FALL 105 EXAMPLE 3 YOUR TURN 3 Find the exact value of the expression without calculator. sin ( π 5 ) csc (9π 5 ) Find the exact value of each expression without calculator. (A) sec ( 37π 18 ) cos ( π 18 ) Find the coterminal angle θ in 0 θ < π of π 5 θ = π 5 + π = 9π 5 Therefore, sin ( π 5 ) = sin (9π 5 ) sin ( 9π 5 ) csc (9π 5 ) = sin ( 9π 5 ) 1 sin ( 9π 5 ) (B) sin (0 ) + cos (380 ) = 1 (C) sin ( 7π 3 ) + cos ( π 3 )

106 DR. YOU: 018 FALL 106 EVEN/ODD FUNCTIONS Even (symmetry with respect to y-axis) cos( θ) = cos(θ) sec( θ) = sec(θ) Odd (symmetry with respect to origin) sin( θ) = sin(θ), csc( θ) = csc(θ) tan( θ) = tan(θ) cot( θ) = cot(θ) EXAMPLE 4 YOUR TURN 4 Find the exact value of the trigonometric function by using even/odd properties Find the exact value of the trigonometric function by using even/odd properties. (A) sin( 45 ) (B) cos( π) (A) sin( 135 ) Since sine is odd, sin( 45 ) = sin(45 ) = Since cosine is even (B) cos ( π 3 ) cos( π) = cos(π) = 1 (C) tan( 10 )

107 DR. YOU: 018 FALL 107 COMPLEMENTARY ANGLES: The complementary angles add up to 90. For example, 70 and 0 are complementary angles, as are 40 and 50. COMPLEMENTARY ANGLE THEOREM: The trig functions can be grouped into three pairs of cofunctions. The cofunction pairs are 1) sine and cosine, ) secant and cosecant, and 3) tangent and cotangent. (Notice that the second function of each cofunction pair is just the first function with the "co-" prefix!) The Complementary Angle Theorem states that cofunctions of complementary angles are equal. So, for example, sin(70 ) = cos(0 ) and cot(40 ) = tan(50 ). EXAMPLE 5 YOUR TURN 5 Find the exact value of each expression. sin(35 ) cos(55 ) Find the exact value of each expression. (A) cos(40 ) sin(50 ) sin(35 ) cos(55 ) = sin(35 ) sin(35 ) = 0 (B) cot(60 ) sin(30 ) sin(60 ) (C) cot(5 ) csc(65 ) sin(5 )

108 DR. YOU: 018 FALL 108 EXAMPLE 6 YOUR TURN 6 Simplify the expression by following the indicated direction. (A) Rewrite in terms of sine and cosine functions: cot θ sec θ Simplify the expression by following the indicated direction. (A) Rewrite over a common denominator 1 1 cos θ cos θ cot θ sec θ = cos θ sin θ 1 cos θ = 1 sin θ (B) Multiply sin θ 1+cos θ 1 cos θ by and simplify 1 cos θ sin θ 1 cos θ 1 + cos θ 1 cos θ = sin θ (1 cos θ) (1 + cos θ)(1 cos θ) (B) Multiply and simplify (cos θ + sin θ) 1 cos θ sin θ = = = sin θ (1 cos θ) 1 cos θ sin θ (1 cos θ) sin θ 1 cos θ sin θ (C) Factor and simplify cos θ 1 cos θ cos θ

109 DR. YOU: 018 FALL 109 EXAMPLE 7 YOUR TURN 7 Establish the identity csc θ cos θ = cot θ Establish the identity sec θ sin θ = tan θ We start from the left side: csc θ cos θ = 1 cos θ sin θ = cos θ sin θ = cot θ EXAMPLE 8 YOUR TURN 8 Establish the identity (1 cos θ)(1 + tan θ) = 1 Establish the identity cos θ (1 + tan θ) = 1 We start from the left side: (1 cos θ)(1 + cot θ) = sin θ csc θ = sin 1 θ sin θ = 1

110 DR. YOU: 018 FALL 110 EXAMPLE 9 YOUR TURN 9 Establish the identity Establish the identity 3 sin θ + 4 cos θ = 3 + cos θ 9 sin θ + 13 cos θ = cos θ We start from the left side 3 sin θ + 4 cos θ = 3(sin θ + cos θ) + cos θ = 3 + cos θ EXAMPLE 10 YOUR TURN 10 Establish the identity Establish the identity (sec θ 1)(sec θ + 1) = tan θ (1 cos θ)(1 + cos θ) = sin θ We start from the left side (sec θ 1)(sec θ + 1) = sec θ + sec θ sec θ 1 : FOIL = sec θ 1 = (tan θ + 1) 1 : Identity = tan θ

111 DR. YOU: 018 FALL 111 EXAMPLE 11 YOUR TURN 11 Establish the identity Establish the identity cos θ (tan θ + cot θ) = csc θ csc θ sin θ = cos θ cot θ We start from the left side and usually we express it as only sine and cosine. cos θ (tan θ + cot θ) sin θ = cos θ ( cos θ cos θ + ) Distributive law sin θ = sin θ + cos θ sin θ multiply sin θ sin θ to sin θ = sin θ sin θ + cos θ sin θ = sin θ + cos θ sin θ make it one fraction identity = 1 sin θ = csc θ

112 DR. YOU: 018 FALL 11 EXAMPLE 1 YOUR TURN 1 Establish the identity sin θ cos θ = 0 cos θ 1 sin θ Establish the identity sin θ cos θ = csc θ cos θ + 1 sin θ We start from the left side and rewrite it as one fraction; sin θ cos θ cos θ 1 sin θ = = sin θ sin θ (cos θ 1)(cos θ + 1) + (cos θ 1) sin θ (cos θ 1) sin θ sin θ (cos θ 1) sin θ + cos θ 1 (cos θ 1) sin θ = sin θ + cos θ 1 (cos θ 1) sin θ = = = (cos θ 1) sin θ 0 (cos θ 1) sin θ

113 DR. YOU: 018 FALL 113 EXAMPLE 13 YOUR TURN 13 Establish the identity 1 sin θ cos θ = cos θ 1 sin θ Establish the identity cos θ = sec θ tan θ 1 + sin θ We know that sin θ + cos θ = 1 cos θ = 1 sin θ cos θ = (1 sin θ)(1 + sin θ) We start from the left side 1 sin θ cos θ = = (1 sin θ)(1 + sin θ) cos θ (1 + sin θ) cos θ cos θ (1 + sin θ) Multiple the conjugate Simplify = cos θ 1 + sin θ

114 DR. YOU: 018 FALL 114 HOMEWORK 9 NAME As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to this coversheet. 1. Use the even-odd properties to find the exact value of each expression. Do not use a calculator. (A) sin( 45 ) (B) cos( 315 ) (C) tan( 5 ) (D) cos ( 4π 3 ). Use the properties of trigonometric functions to find the exact value of each expression. Do not use a calculator. (A) sin (67 ) + cos (67 ) (C) tan(89 ) sin(89 ) cos(89 ) (B) sin(3 ) csc(3 ) (D) cos ( 5π ) sec (19π ) Establish each identity (A) (sec x 1)(sec x + 1) = tan x (C) (sec x tan x) = 1 sin x 1+sin x (B) 3 sin x + 4 cos x = 3 + cos x (D) cos x 1+sin x + 1+sin x cos x = sec x 4. Simplify: (A) (sin x+cos x) 1+ sin x cos x (B) sin x 1 sin x sin x

115 DR. YOU: 018 FALL 115 LECTURE -5 SOLVING A EQUATIONS. How to find all solutions of a simple trigonometric equation Find all solution on [0,π) of a trigonometric equation Find all solution on [0,π) of a trigonometric equation by factoring 1) Case 1 ) Case 3) Case 3 EXAMPLE 1 YOUR TURN 1 Find all solutions of Find all solutions of sin θ = 1 sin θ = 3 sin θ = 1 1) The period is π b = π. ) Find all θ in unit circle such that sin θ = 1. Then θ = π 6 and 5π 6. Since this sine function has period π, the solutions are θ = π + πk or 6 θ = 5π 6 + πk for any integer k

116 DR. YOU: 018 FALL 116 EXAMPLE YOUR TURN Find all solutions of Find all solutions of cos θ = cos θ = 1 1) The period is π b = π. ) Find all θ in unit circle such that cos θ = Then θ = 3π 4 and 5π 4. Since this cosine function has period π, the solutions are θ = 3π 4 θ = 5π 4 + πk or + πk for any integer k EXAMPLE 3 YOUR TURN 3 Find all solutions of Find all solutions of tan θ = 3 tan θ = 1 1) The period is π b = π. ) Find all θ in unit circle such that tan θ = 3 and 0 θ π. θ = π 3 Since this tangent function has period π, the solutions are θ = π + πk for any integer k 3

117 DR. YOU: 018 FALL 117 EXAMPLE 4 YOUR TURN 4 Find all solutions of Find all the solutions of sin(3θ) = 1 sin ( θ ) = 3 1) Since sin(3θ) = 1 π, its period is = π. b 3 ) Find angle 3θ in unit circle such that sin(3θ) = 1. Then 3θ = π θ = π 18 6 and 3θ = 5π 6 5π and θ = 18 Since this sine function has period π, the solutions are 3 θ = π 18 + π 3 k or θ = 5π 18 + π 3 k for any integer k EXAMPLE 5 YOUR TURN 5 Find all solutions of Find all solutions of tan ( θ ) = 1 tan(θ) = 1 1) Since tan ( 1 π θ) = 1, its period is = π = π. b 1/ ) Find angle 3θ in unit circle on [0, π] such that tan ( θ ) = 1. Then θ = π 4 θ = π Since this tangent function has period π, the solutions are θ = π + πk for any integer k

118 DR. YOU: 018 FALL 118 EXAMPLE 6 YOUR TURN 6 Solve the equation on the interval 0 θ < π Solve the equation on the interval 0 θ < π sin θ = 3 cos θ = 1 1) sin θ = 3 and its period is π (the graph of sin(θ) on 0 θ < π has 1 cycle) ) Find all θ in unit circle such that sin θ = 3. θ = 4π 3 and 5π 3 Then, the solution set is { 4π 3, 5π 3 } EXAMPLE 7 YOUR TURN 7 Solve the equation on the interval 0 θ < π. tan θ = 3 3 Solve the equation on the interval 0 θ < π. tan θ = 1 1) tan θ = 3 3 and its period is π (the graph of tan(θ) on 0 θ < π has cycles) ) Find all θ on 0 x π such that tan θ = 3 3 θ = π 6 3) Since its period is π, θ = π 6 + π = 7π 6 is also a solution. Then, the solution set is { π 6, 7π 6 }

119 DR. YOU: 018 FALL 119 EXAMPLE 8 YOUR TURN 8 Solve the equation on the interval 0 θ < π. 4 sin θ 1 = 0 Solve the equation on the interval 0 θ < π. 4 cos θ 3 = 0 First, find sin θ by using quadratic formula. 0 ± 0 4(4)( 1) sin θ = (4) = 0 ± 4 8 sin θ = 1, sin θ = 1 The period of sin(θ) is π. (the graph of sin(θ) on 0 θ < π has 1 cycle) Second, find all θ in unit circle such that sin θ = ± 1. θ = π 6, 5π 6 θ = 7π 6, 11π 6 when sin θ = 1 when sin θ = 1 The solution set is { π 6, 5π 6, 7π 6, 11π 6 }

120 DR. YOU: 018 FALL 10 EXAMPLE 9 YOUR TURN 9 Solve the equation on the interval 0 θ < π. Solve the equation on the interval 0 θ < π. sin θ cos θ = cos θ cos θ = cos θ Make one side zero; sin θ cos θ cos θ = 0 Factor out cos θ cos θ ( sin θ 1) = 0 cos θ = 0 or sin θ 1 = 0 cos θ = 0 or sin θ = 1 The periods of sin(θ), cos(θ) are π. (the graphs of sin(θ), cos(θ) on 0 θ < π have 1 cycle) Find all θ in unit circle such that cos θ = 0 or sin θ = 1 θ = π, 3π when cos θ = 0 θ = π 6, 5π 6 when sin θ = 1 The solution set is { π, 3π, π 6, 5π 6 }

121 DR. YOU: 018 FALL 11 EXAMPLE 10 YOUR TURN 10 Solve the equation on the interval 0 θ < π Solve the equation on the interval 0 θ < π. sin θ 3 sin θ + 1 = 0 cos θ + cos θ 1 = 0 First, find sin θ by using quadratic formula. ( 3) ± ( 3) 4()(1) sin θ = () = 3 ± 1 4 sin θ = 1 or sin θ = 1 The period of sin(θ) is π. (the graph of sin(θ) on 0 θ < π has 1 cycles) Second, find all θ in unit circle such that sin θ = 1, 1 θ = π 6, 5π 6 when sin θ = 1 θ = π when sin θ = 1 The solution set is { π 6, 5π 6, π }

122 DR. YOU: 018 FALL 1 EXAMPLE 11 YOUR TURN 11 Solve the equation on the interval 0 θ < π Solve the equation on the interval 0 θ < π cos θ + sin θ 1 = 0 sin θ = 3 cos θ First, use the identity (1 sin θ) + sin θ 1 = 0 sin θ + sin θ 1 = 0 sin θ + sin θ + 1 = 0 Second, find sin θ by using quadratic formula. 1 ± (1) 4( )(1) sin θ = ( ) = 1 ± 3 4 sin θ = 4, 4 4 sin θ = 1, 1 The period of sin(θ) is π. (the graph of sin(θ) on 0 θ < π has 1 cycles) Second, find all θ in unit circle such that sin θ = 1, 1 θ = 7π 6, 11π 6 when sin θ = 1 θ = π when sin θ = 1 The solution set is { 7π 6, 11π 6, π }

123 DR. YOU: 018 FALL 13 EXAMPLE 1 YOUR TURN 1 Solve the equation on the interval 0 θ < π Solve the equation on the interval 0 θ < π sin (θ π 3 ) = 1 tan (θ + π 6 ) = 1 The period of sin (θ π π ) is = π. 3 b (the graph of sin(θ π ) on 0 θ < π has 1 cycles) 3 Find all θ π 3 in unit circle such that the value of sine is 1 θ π 3 = π 6 θ = π 6 + π 3 = π θ π 3 = 5π 6 θ = 5π 6 + π 3 = 7π 6 The solution set is { π, 7π 6 }

124 DR. YOU: 018 FALL 14 EXAMPLE 13 YOUR TURN 13 Solve the equation on the interval 0 θ < π. Solve the equation on the interval 0 θ < π. sin(4θ) = 1 cos(θ) = 1 Its period is π 4 = π (the graph of sin(4θ) on 0 θ < π has 4 cycles) Find all 4θ in unit circle such that sin(4θ) = 1. 4θ = 7π 6 θ = 7π 4 11π or 4θ = 6 and 11π 4 which lies in [0, π ) Then there are eight solutions; θ = 7π 4, 7π 4 + π = 19π 4, 7π 4 + π = 31π 4, 7π 4 + 3π = 43π 4 θ = 11π 4, 11π 4 + π = 3π 4, 11π 4 + π = 35π 4, 11π 4 + 3π = 47π 4

125 DR. YOU: 018 FALL 15 HOMEWORK 10 NAME As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to this coversheet. 1. Find all solutions of the equation (A) sin θ = (B) tan θ = 3 (C) cos θ 1 = 0 (D) sin(θ) = 1. Find the solutions of the equation that are in the interval [0,π) (A) 4 sin θ 1 = 0 (B) ( sin θ 1)( cos θ + 3) = 0 (C) tan θ sin θ = sin θ (D) sin θ = 1 sin θ (E) cos θ = cos θ (F) cos θ + sin θ = 1

126 DR. YOU: 018 FALL 16 REVIEW PROBLEMS FOR EXAM Click the blue words to see the video 1. Evaluate csc, sec, cot by using calculator: Problem 1. Evaluating the inverse of trigonometric functions without a calculator : Problem 3. Evaluate the composition of trigonometric functions: inverse trig (trig) : Problem 3, 4 4. Evaluate the composition of trigonometric functions: Case : trig (inverse trig) : Problem 5,6 5. How to find all solutions of a simple trigonometric equation 6. Find all solution on [0,π) of a trigonometric equation : Problem 9(A), 9(B) 7. Find all solution on [0,π) of a trigonometric equation by factoring: Problem 9(C), 9(D) 1) Case 1 ) Case 3) Case 3 8. Amplitude, phase shift, period of sine/cosine: Problem Domain, Range, Period of sec, csc, cot : Problem 11,1 10. Phase shift, period, Vertical asymptote of tangent : Problem Find the sine function given conditions: Problem How to draw the graph of cosine: Problem Find the approximate degree value for cot 1 ( ). Round your answer to nearest hundredth of a degree. Find the exact value of tan 1 ( 3) in radian without a calculator. 3. Find the exact value of cos 1 (cos ( 3π )) in radian without a calculator Find the exact value of sin 1 (sin ( 9π )) in radian without a calculator Find the exact value of sec (tan 1 ( 1 )) without a calculator Find the exact value of sin (cos 1 ( 1 )) without a calculator Simplify the expression by using identity: (sin θ+cos θ) 1 sin θ cos θ 8. Establish the identity: (A) csc θ sin θ = cos θ cot θ (B) (sec x 1) cot x = sin x tan x sin x+cos x

127 DR. YOU: 018 FALL Find all solution θ of the equations in the interval [0,π). (A) sin θ + 1 = 0 (B) cos(θ) = 1 (C) sin θ = sin θ (D) sin θ + 3 cos θ = Find the amplitude, period, phase shift, domain, and range of the function (A) y = 5 cos(πx 5) (B) y = 6 sin ( x π 5 ) 11. Find the period, phase shift, range of the function: y = 4 cot(x) 1. Find the period, range, phase shift of the function: y = 5 sec ( x 3 π) 13. Find all values of x, π x π, in which the graph of the function f(x) = tan x have vertical asymptote. 14. Find y = A sin(bx + c) + d such that it has the following properties; Amplitude: 3 Period: π Phase shift: π 15. Graph the function f(x) = cos (3x + π ) using a scale of π on the x-axis and 1 on the y-axis. 3

128 DR. YOU: 018 FALL 18 SOLUTIONS 1. cot 1 ( ) = tan 1 1 ( ) = θ = tan 1 ( 3) is an angle which is in π θ π. In unit circle, θ = π cos 1 (cos ( 3π 5 )) = π cos 1 (cos( 108 )) = 108 = = 3π 5 sin 1 (sin ( 9π 8 )) = π sin 1 (sin(0.5 )) =.5 = = π 8 5. We have to find x, y, r such that θ = tan 1 ( 1 3 ), tan θ = 1 3 = y x and π θ π Then θ lies in I quadrant, x = 3, y = 1, r = (1) + (3) = 10 sec (tan 1 ( 1 3 )) = r x = We have to find x, y, r such that θ = cos 1 ( 1 1 ), cos θ = = x r and 0 θ π Then θ lies in II quadrant, x = 1, r = 13, y = 5 since ( 1) + y = (13) sin (cos 1 ( 1 13 )) = y r = (sin θ+cos θ) 1 sin θ cos θ 8. = sin θ+ sin θ cos θ+cos θ 1 sin θ cos θ = 1+ sin θ cos θ 1 sin θ cos θ = sin θ cos θ sin θ cos θ = (A) csc θ sin θ = 1 1 sin θ = sin θ = 1 sin θ = cos θ = cos θ cos θ = cos θ cot θ sin θ sin θ sin θ sin θ sin θ sin θ (B) (sec x 1) cot x tan x sin x+cos x = 9. (A) sin θ = 1 In unit circle, θ = 7π 6, 11π 6 (B) cos(θ) = 1 tan x cot x tan x sin x+cos x = tan x tan x sin x+cos x = sin x cos x sin x cos x sin = sin x+cos x x sin x+cos x and its period of sin θ is π ( the graph has one cycle in [0,π)) and its period of cos(θ) is π (the graph has two cycles in [0,π)) = sin x In unit circle, θ = π 3, 4π 3 θ = π 3, π 3 Since it has two cycles, θ = π 3 + π = 4π 3, π 3 + π = 5π 3 are also solutions. Then the solution set is { π 3, π 3, 3π 3, 5π 3 } (C) sin θ = sin θ sin θ sin θ = 0 sin θ ( sin θ 1) = 0 sin θ = 0, sin θ = 1 Its period of sin θ is π (the graph has one cycle in [0,π)) In unit circle, θ = 0, π when sin θ = 0 ; θ = π 6, 5π 6 when sin θ = 1 The solution set is {0, π, π 6, 5π 6 } (D) sin θ + 3 cos θ = 0

129 DR. YOU: 018 FALL 19 (1 cos 1 θ) + 3 cos θ = 0 cos 1 θ + 3 cos θ = 0 0 = cos 1 θ 3 cos θ By quadratic formula cos θ = ( 3) ± ( 3) 4()( ) () = 3 ± 5 4 Since 1 cos θ 1, cos θ = 1 is only possible =, 1 Its period of cos(θ) is π (the graph has one cycle in [0,π)) In unit circle, θ = π 3, 4π (A) Amplitude:5 period:1 Phase shift: 5 (B) Amplitude:6 period:4π Phase shift: π Period: π, phase shift:0, Range: (, ) π Domain: (, ) Domain: (, ) Range: [ 5,5] Range: [ 6,6] 1. Period: 6π, phase shift:3π, Range: (, 5] [5, ) 13. x = 3π, x = π, x = π, x = 3π 14. y = 3 sin(x π) or y = 3 sin(x π) 15. Find the amplitude, Period, and Phase shift. Amplitude: = Period: π b = π (3) = π 3 Phase shift: c b = π 3 = π 6 To find the five key points of y = cos (3x + π period ), find t = = π/3 = π Five x-values: 0, π, π, 3π, 4π Find the corresponding y-values: it starts.:, 0,, 0, Vertical shift:0 (0, ) ( π 6, 0) (π 6, ) (3π 6, 0) (4π 6, ) Five points in y = cos (3x + π ) : add (phase shift, vertical shift) ( π 6, ) (0, 0) (π 6, ) (π 6, 0) (3π 6, ) 4 y = cos 3 x + π ( ) π π 4

130 DR. YOU: 018 FALL 130 FORMULAS SHEET Fundamental Identities tan θ = sin θ cos θ cot θ = cos θ sin θ cot θ = 1 tan θ csc θ = 1 sin θ sec θ = 1 cos θ Half angle Formulas Double angle Formulas Sum and difference Formulas Sum to product Formulas Product to Sum Formulas Solving Triangles Complex numbers Vector sin θ + cos θ = 1 tan θ + 1 = sec θ cot θ + 1 = csc θ sin ( θ ) = ± 1 cos θ sin(θ) = sin θ cos θ cos ( θ ) = ± 1 + cos θ (the sign is determined by the quadrant of θ ) tan θ tan(θ) = 1 tan θ tan ( θ 1 cos θ ) = sin θ cos(θ) = cos θ sin θ cos(θ) = cos θ 1 cos(θ) = 1 sin θ sin(a + b) = sin a cos b + cos a sin b sin(a b) = sin a cos b cos a sin b cos(a + b) = cos a cos b sin a sin b cos(a b) = cos a cos b + sin a sin b tan(a + b) = tan(a b) = tan a + tan b 1 tan a tan b tan a tan b 1 + tan a tan b a + b b + b b sin a + sin b = sin ( ) cos (a ) cos a + cos b = cos (a ) cos (a ) a b + b + b b sin a sin b = sin ( ) cos (a ) cos a cos b = sin (a ) sin (a ) sin a sin b = 1 {cos(a b) cos(a + b)} cos sin a cos b = 1 {sin(a + b) + sin(a b)} a cos b = 1 {cos(a b) + cos(a + b)} Area = 1 a+b+c ab sin(c) or Area = s(s a)(s b)(s c), s = LAW OF Sines sin(a) a = sin(b) b = sin(c) c LAW of Cosines Let z 1 = r 1 (cos(a) + i sin(a)) and z = r (cos(b) + i sin(b)) z 1 z = r 1 r (cos(a + b) + i sin(a + b)) z 1 z = r 1 r (cos(a b) + i sin(a b)) Let z = a + bi be a complex number The conjugate of z is z = a bi z = a + b Let v = a 1 i + b 1 j and w = a i + b j c = a + b ab cos(c) Unit vector in same direction as v is u = v v, v = a 1 + b 1 Dot product: v w = a 1 a + b 1 b If θ is between vectors v and w: cos θ = v w v w

131 DR. YOU: 018 FALL 131 CHAPTER 3. FORMULAS & APPLICATIONS LECTURE 3-1 TRIGONOMETRY IDENTITIES AND SUM/DIFFERENCE FORMULA Sum and difference Formulas sin(a + b) = sin a cos b + cos a sin b sin(a b) = sin a cos b cos a sin b cos(a + b) = cos a cos b sin a sin b cos(a b) = cos a cos b + sin a sin b tan(a + b) = tan(a b) = tan a + tan b 1 tan a tan b tan a tan b 1 + tan a tan b Evaluate the exact value by using sum and difference formula : Click to see video 1) Case 1 ) Case 3) Case 3 4) Case 4 EXAMPLE 1 YOUR TURN 1 Find the exact value of each expression. (A) sin(0 ) cos(10 ) + cos(0 ) sin(10 ) Find the exact value of each expression. (A) sin(40 ) cos(50 ) + cos(40 ) sin(50 ) (B) cos(80 ) sin(0 ) sin(80 ) cos(0 ) sin(0 ) cos(10 ) + cos(0 ) sin(10 ) = sin( ) = sin(30 ) (B) sin(79 ) cos(34 ) cos(79 ) sin(34 ) = 1 cos(80 ) sin(0 ) sin(80 ) cos(0 ) = sin(0 ) cos(80 ) cos(0 ) sin(80 ) = sin(0 80 ) (C) sin ( 3π 8 ) cos (π 8 ) + cos (3π 8 ) sin (π 8 ) = sin( 60 ) = 3

132 DR. YOU: 018 FALL 13 EXAMPLE YOUR TURN Find the exact value of each expression. Find the exact value of each expression. (A) cos(70 ) cos(0 ) + sin(0 ) sin(0 ) (A) cos(75 ) cos(15 ) + sin(75 ) sin(15 ) (B) cos(50 ) cos(10 ) sin(50 ) sin(10 ) cos(70 ) cos(5 ) + sin(70 ) sin(5 ) = cos(70 5 ) = cos(45 ) = (B) cos ( π 1 ) cos (π 4 ) sin ( π 1 ) sin (π 4 ) cos(50 ) cos(10 ) sin(50 ) sin(10 ) = cos( ) = cos(60 ) = 1 EXAMPLE 3 YOUR TURN 3 Find the exact value of each expression. tan(0 ) + tan(5 ) 1 tan(0 ) tan(5 ) Find the exact value of each expression. (A) tan(40 ) tan(10 ) 1+tan(40 ) tan(10 ) tan(0 ) + tan(5 ) 1 tan(0 ) tan(5 ) = tan(0 + 5 ) = tan(45 ) = 1 (B) tan(5π 1 )+tan(π 4 ) 1 tan( 5π 1 ) tan(π 4 )

133 DR. YOU: 018 FALL 133 EXAMPLE 4 YOUR TURN 4 Write the expression in terms of a single trigonometric Write the expression in terms of a single trigonometric function. function. sin(3x) cos(4x) + cos(3x) sin(4x) (A) cos(3x) cos(8x) sin(3x) sin(8x) sin(3x) cos(4x) + cos(3x) sin(4x) = sin(3x + 4x) = sin(7x) (B) 6 sin(11x) cos(5x) 6 cos(11x) sin(5x) EXAMPLE 5 YOUR TURN 5 Express the following angle as sum/difference of two Express the following angle as sum/difference of two special angles. special angles. (A) 195 = (A) 75 (B) π 1 = 15 = (B) 11π 1 (C) 85

134 DR. YOU: 018 FALL 134 EXAMPLE 5 YOUR TURN 5 Find the exact value of Find the exact value of sin(105 ) sin(75 ) sin(105 ) = sin( ) = sin(60 ) cos(45 ) + cos(60 ) sin(45 ) = = EXAMPLE 6 YOUR TURN 6 Find the exact value of cos ( 19π 1 ) Find the exact value of cos ( 7π 1 ) cos ( 19π 1 ) = cos(85 ) = cos( ) = cos(40 ) cos(45 ) sin(40 ) sin(45 ) = ( 1 ) = ( )

135 DR. YOU: 018 FALL 135 EXAMPLE 6 YOUR TURN 6 Find the exact value of cos ( 7π 1 ) Find the exact value of sin(55 ) cos ( 7π 1 ) = cos(105 ) = cos( ) = cos(60 ) cos(45 ) sin(60 ) sin(45 ) = ( 1 ) ( 3 ) = 6 4 YOUR TURN YOUR TURN Find the exact value of sin ( 3π 1 ) Find the exact value of cos(195 )

136 DR. YOU: 018 FALL 136 EXAMPLE 7 YOUR TURN 7 Find the exact value of Find the exact value of tan( 15 ) tan(75 ) tan( 15 ) = tan(30 45 ) = = = tan(30 ) tan(45 ) 1 + tan(30 ) tan(45 ) ( 3 3 1) 3 ( ) 3 = ( 3 3)(3 3) = (3 + 3)(3 3) = = = ( + 3) 6 = + 3

137 DR. YOU: 018 FALL 137 EXAMPLE 8 YOUR TURN 8 Find the exact value of sin(a + b) when sin a = 3 5, 0 < a < π and cos b = 1 13, π < b < 0 Find the exact value of sin(a b) when cos a = 4 5, 0 < a < π 1 and sin b =, π < b < We find points on terminal side which is corresponding to angles a and b. quadrant x y r a : I b : IV sin(a + b) = sin a cos b + cos a sin b = ( 3 5 ) (1 13 ) + (4 5 ) ( 5 13 ) = 16 65

138 DR. YOU: 018 FALL 138 EXAMPLE 9 YOUR TURN 9 Find the exact value of cos(a + b) when tan a = 4 3, π < a < π and cos b = 7 5, 0 < b < π. Find the exact value of cos(a b) when sin a = 5 13, π < a < 0 and cos b = 3 5, 0 < b < π We find points on terminal side which is corresponding to angles a and b. quadrant x y r a : II b : I cos(a + b) = cos a cos b sin a sin b = ( 3 5 ) ( 7 5 ) + (4 5 ) (4 5 ) = = 3 5

139 DR. YOU: 018 FALL 139 YOUR TURN Find the exact value of sin(a b) when cos a = 8 17, π < a < 0 and sin b = 3 5, π < b < 3π. YOUR TURN Find the exact value of tan(a + b) when sin a = 3, π < a < 3π and cos b = 7, 3π < b < π. 5 5

140 DR. YOU: 018 FALL 140 EXAMPLE 10 YOUR TURN 10 Verify the identity Verify the identity sin ( π + θ) = cos θ cos(π θ) = cos θ sin ( π + θ) = sin ( π ) cos θ + cos (π ) sin θ = 1 cos θ + 0 sin θ = cos θ

141 DR. YOU: 018 FALL 141 EXAMPLE 11 YOUR TURN 11 Find the exact value of Find the exact value of sin [cos 1 ( 1 ) + sin 1 ( 3 5 )] cos [tan 1 ( 4 3 ) + cos 1 ( 5 13 )] Let a = cos 1 ( 1 ) and b = sin 1 ( 3 5 ). Then cos a = + 1, 0 a 180 sin b = 3, 90 b 90 5 We find points on terminal side which is corresponding to angles a and b. quadrant x y r a : I 1 3 b : IV sin [cos 1 ( 1 ) + sin 1 ( 3 5 )] = sin(a + b) = sin a cos b + cos a sin b = ( 3 ) (4 5 ) + (1 ) ( 3 5 ) =

142 DR. YOU: 018 FALL 14 EXAMPLE 1 YOUR TURN 1 Find the exact value of cos [tan 1 ( 4 3 ) + cos 1 ( 5 13 )] Find the exact value of tan [sin 1 ( 1 13 ) + cos 1 ( 4 5 )] Let a = tan 1 ( 4 3 ) and b = cos 1 ( 5 13 ). Then tan a = + 4, 90 b 90 3 cos b = 5, 0 a We find points on terminal side which is corresponding to angles a and b. quadrant x y r a : I b : II cos [tan 1 ( 4 3 ) + cos 1 ( 5 13 )] = cos(a + b) = cos a cos b sin a sin b = ( 3 5 ) ( 5 13 ) + (4 5 ) (1 13 ) = 33 65

143 DR. YOU: 018 FALL 143 NOTE: (Cofunction Identities) Trig identities showing the relationship between sine and cosine, tangent and cotangent, and secant and cosecant. The value of a trig function of an angle equals the value of the cofunction of the complement of the angle. sin(90 θ) = cos θ cos(90 θ) = sin θ tan(90 θ) = cot θ cot(90 θ) = tan θ sec(90 θ) = csc θ csc(90 θ) = sec θ

144 DR. YOU: 018 FALL 144 HOMEWORK 11 NAME As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to this coversheet. 1. Express as a trigonometric function (A) sin(70 ) cos(53 ) + cos(70 ) sin(53 ) (C) cos(3x) cos(4x) + sin(3x) sin(4x) (B) sin(5x) cos(3x) cos(5x) sin(3x) (D) tan(5x) tan(x) 1+tan(5x) tan(x). Find the exact values by using sum and difference formula. (A) cos ( 5π ) (B) sin (11π ) 1 1 (C) tan(85 ) (D) sin (sin 1 ( 5 13 ) cos 1 ( 4 5 )) 3. If sin a = 4 and sec b = 5 such that a is in III quadrant and b is in I quadrant. 5 3 (A) sin(a + b) (B) tan(a + b) 4. If tan a = 7 and cot b = 4 such that a is in II quadrant and b is in III quadrant. 4 3 (A) sin(a b) (B) cos(a b) 5. Find the exact value of cos ( π 3 + θ) when sin θ = 1 3 and 0 < θ < π. 6. Verify the identity: cos (x + 3π ) = sin x 7. (extra) Find the exact value of cos(x y) when sin x + sin y = 1 and cos x + cos y = (extra) Find all solutions, x, in [0, π] of the equation sin (x π ) = cos (x).

145 DR. YOU: 018 FALL 145 LECTURE 3- HALF/DOUBLE AND PRODUCT/SUM FORMULA Double angle Formulas Half angle Formulas sin(θ) = sin θ cos θ tan(θ) = tan θ 1 tan θ cos(θ) = cos θ sin θ cos(θ) = cos θ 1 cos(θ) = 1 sin θ sin ( θ ) = ± 1 cos θ cos ( θ ) = ± 1 + cos θ (the sign is determined by the quadrant of θ ) tan ( θ 1 cos θ ) = sin θ Evaluate the exact value of trigonometry using the double angle formulas: Click to see the video Evaluate the exact value of trigonometry using the half angle formulas: 1) Case 1 ) Case 3) Case 3 Evaluate the exact value of trigonometry with inverse trig EXAMPLE 1 YOUR TURN 1 Find the exact value of cos (75 ) sin (75 ) Find the exact value of (A) cos (105 ) sin (105 ) cos (75 ) sin (75 ) = cos( 75 ) = cos(150 ) = 3 (B) cos (15 ) 1 (C) 1 sin (.5 )

146 DR. YOU: 018 FALL 146 EXAMPLE YOUR TURN Find the exact value of sin(15 ) cos(15 ) Find the exact value of (A) sin(.5 ) cos(.5 ) sin(15 ) cos(15 ) = sin( 15 ) = sin(30 ) = 1 (B) sin(15 ) cos(15 ) EXAMPLE 3 YOUR TURN 3 Find the exact value of sin(.5 ) by half-angle Find the exact value of cos(15 ) by half-angle formula. formula. Since.5 is in IV quadrant, the value of sine is negative; sin(. 5 ) = sin ( 45 ) half angle formula = 1 cos( 45 ) negative = 1 = 4 =

147 DR. YOU: 018 FALL 147 EXAMPLE 4 YOUR TURN 4 Find the exact value of sin ( π ) by half-angle formula. 8 Find the exact value of cos ( 5π ) by half-angle formula. 1 Since π =.5 is in I quadrant, the value of sine is 8 positive; sin ( π 8 ) = sin(.5 ) = sin ( 45 ) half angle formula = + 1 cos(45 ) positive = 1 = 4 = EXAMPLE 5 YOUR TURN 5 Find the exact value of tan(15 ) by half-angle formula. Find the exact value of tan ( 9π ) by half-angle formula. 4 tan(15 ) = tan ( 30 ) half angle formula = 1 cos(30 ) sin(30 ) = = 3 1 = 3

148 DR. YOU: 018 FALL 148 EXAMPLE 6 YOUR TURN 6 Find the exact value of cos(θ) if Find the exact value of cos(θ) if sin(θ) = 5 13 (A) cos(θ) = 3 5 cos(θ) = 1 sin θ = 1 ( 5 13 ) = (B) sin(θ) = 3 7 (C) cos(θ) = 1 3

149 DR. YOU: 018 FALL 149 EXAMPLE 7 YOUR TURN 7 If sin θ = 3 5 and π < θ < 3π, find If sin θ = 4 5 and π < θ < π, find (A) sin(θ) (B) cos(θ) (A) sin(θ) (B) cos(θ) (C) sin ( θ ) (D) cos (θ ) (C) sin ( θ ) (D) cos (θ ) First, find the terminal point of the angle θ and radius: It is in III quadrant and sin θ = 3 5 ( 4, 3), r = 5 (A) sin(θ) = sin θ cos θ = ( 3 5 ) ( 4 5 ) = 4 5 (B) cos(θ) = 1 sin θ = 1 ( 3 5 ) = 7 5 (C) π θ 3π 4 is in II quadrant and sin (θ ) > 0 sin ( θ ) = + 1 cos θ = 1 ( 4 5 ) = 9 10 = (D) π θ 3π 4 is in II quadrant and cos (θ ) < 0 cos ( θ ) = 1 + cos θ = 1 + ( 4 5 ) = 1 10 = 10 10

150 DR. YOU: 018 FALL 150 YOUR TURN If sin θ = 1 13 and π < θ < 3π, find (A) sin(θ) (B) cos(θ) (C) sin ( θ ) (D) cos (θ ) YOUR TURN If tan θ = 15 8 and 0 < θ < π, find (A) sin(θ) (B) cos(θ) (C) sin ( θ ) (D) cos (θ )

151 DR. YOU: 018 FALL 151 EXAMPLE 8 YOUR TURN 8 Find the exact value of Find the exact value of (A) cos [ 1 sin 1 ( 4 5 )] (A) cos [ sin 1 ( 5 13 )] = cos [ sin 1 ( 4 5 ) ] 1+cos θ = Since θ = sin 1 ( 4 5 ) 90 < θ < 0 45 < θ < 0 ; x = 3, y = 4, r = 5 = (B) sin [ 1 cos 1 ( 5 13 )] = 4 5 = 5 = 5 5 (B) sin ( cos 1 ( 3 )) 4 Since θ = cos 1 ( 3 ) 5 0 < θ < 90 ; x = 3, y = 4, r = 5 = sin θ cos θ = = 4 5

152 DR. YOU: 018 FALL 15 Sum to product Formulas Product to Sum Formulas a + b b + b b sin a + sin b = sin ( ) cos (a ) cos a + cos b = cos (a ) cos (a ) a b + b + b b sin a sin b = sin ( ) cos (a ) cos a cos b = sin (a ) sin (a ) sin a sin b = 1 {cos(a b) cos(a + b)} cos sin a cos b = 1 {sin(a + b) + sin(a b)} a cos b = 1 {cos(a b) + cos(a + b)} Evaluate the exact value of trigonometry by using sum to product formulas Evaluate the exact value of trigonometry by using product to sum formulas EXAMPLE 9 YOUR TURN 9 Find the exact value of sin(75 ) + sin(15 ) Find the exact value of cos(55 ) cos(165 ) sin(75 ) + sin(15 ) = sin ( ) cos ( ) = sin(45 ) cos(30 ) = ( ) (1 ) = EXAMPLE 10 YOUR TURN 10 Find the exact value of sin(195 ) cos(75 ) Find the exact value of cos(55 ) cos(15 ) sin(195 ) cos(75 ) = 1 {sin( ) + sin( )} = 1 {sin(70 ) + sin(10 )} = 1 3 { 1 + } = + 3 4

153 DR. YOU: 018 FALL 153 EXAMPLE 11 YOUR TURN 11 Simplify the expression by using sum to product formula Simplify the expression by using sum to product formula 6 sin(x) 6 sin(4x) cos(5x) cos(x) 6 sin(x) 6 sin(4x) = 6{sin(x) sin(4x)} x 4x x + 4x = 6 sin ( ) cos ( ) = 1 sin( x) cos(3x) odd = 1 sin(x) cos(3x) YOUR TURN Simplify the expression by using sum to product formula sin(3x) + sin(x) YOUR TURN Simplify the expression by using sum to product formula 3 cos(x) + 3 cos(5x)

154 DR. YOU: 018 FALL 154 EXAMPLE 1 YOUR TURN 1 Simplify the expression by using product to sum formula 4 cos(10x) sin(6x) Simplify the expression by using product to sum formula sin(6x) cos(x) 4 cos(10x) sin(6x) = 4 sin(6x) cos(10x) = 4 1 {sin(6x + 10x) + sin(6x 10x)} = {sin(16x) + sin( 4x) } odd = {sin(16x) sin(4x)} YOUR TURN Simplify the expression by using product to sum formula cos(3x) cos(5x) YOUR TURN Simplify the expression by using product to sum formula sin(4x) sin(x)

155 DR. YOU: 018 FALL 155 EXAMPLE 13 YOUR TURN 13 Solve the equation on the interval 0 θ < π. cos(θ) = cos θ Solve the equation on the interval 0 θ < π. sin(θ) = sin θ By the double formula, cos(θ) = cos θ cos θ 1 = cos θ cos θ cos θ 1 = 0 Use the quadratic formula to find cos θ cos θ = ( 1) ± ( 1) 4()( 1) () = 1 ± 9 4 = 1 ± 3 4 = 1, 1 The period of cos(θ) is π. (the graph of cos(θ) on 0 θ < π has 1 cycles) Find all θ in unit circle such that cos θ = 1, cos θ = 1 θ = π 3, 4π 3 when cos θ = 1 θ = 0, π when cos θ = 1 Then the solution set is { π 3, 4π 3, 0, π}

156 DR. YOU: 018 FALL 156 EXAMPLE 14 YOUR TURN 14 Solve the equation on the interval 0 θ < π. sin(θ) + sin(4θ) = 0 Solve the equation on the interval 0 θ < π. sin(θ) sin(3θ) = 0 By the sum to the product formula 0 = sin(θ) + sin(4θ) θ + 4θ θ 4θ 0 = sin ( ) cos ( ) 0 = sin(3θ) cos( θ) 0 = sin(3θ) cos(θ) sin(3θ) = 0, cos(θ) = 0 The period of sin(3θ) is π. 3 (the graph of sin(3θ) on 0 θ < π has 3 cycles) All 3θ in unit circle such that sin(3θ) = 0 are 3θ = 0 θ = 0 3θ = π θ = π 3 Since its period is π, the solutions are 3 θ = 0, 0 + π 3, 0 + (π 3 ) = 4π 3 θ = π 3, π 3 + π 3 = π, π 3 + (π 3 ) = 5π 3 The period of cos(θ) is π. (the graph of cos(θ) on 0 θ < π has 1 cycles) All θ in unit circle such that cos θ = 0 are θ = π, 3π Then, the solution set is {0, π 3, 4π 3, π 3, π, 5π 3, π, 3π }

157 DR. YOU: 018 FALL 157 HOMEWORK 1 NAME As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to this coversheet. 1. If cos θ = 3 and 0 < θ < 90, find 5 (A) sin(θ) (B) cos(θ) (C) tan(θ) (D) sin ( θ ) (E) cos (θ) (F) tan (θ). If csc θ = 5 3 and 90 < θ < 0, find (A) sin(θ) (B) cos(θ) (C) tan(θ) (D) sin ( θ ) (E) cos (θ ) (F) tan (θ ) 3. Use the half-angle formula to find the exact values of sin ( 3π 8 ). 4. Use the half-angle formula to find the exact values of cos(165 ). 5. Find the exact value of cos(θ) when sin(θ) = Express as a sum. (A) sin(7x) sin(x) (B) sin(4x) cos(8x) 7. Express as a product (A) sin(x) + sin(4x) (B) cos(5x) cos(3x) 8. (extra) Find the exact value of sin ( tan 1 ( 5 1 )) 9. (extra) Find the exact value of cos ( 1 tan 1 ( 8 15 )) 10. (extra) Find the exact value of sin(50 )+sin(10 ) cos(50 )+cos(10 ). 11. (extra) Find the exact value of tan(θ) if sin ( θ ) = (extra) Find all solutions, x, in [0,π) of the equation sin(x) = cos x cos x.

158 DR. YOU: 018 FALL 158 LECTURE 3-3 AREA OF A TRIANGLE AREA OF A TRIANGLE SAS case: c h B a Area of a triangle is 1 ab sin(c) or 1 bc sin(a) or 1 ca sin(b) A b C SSS case (Heron s formula): Area of a triangle is s(s a)(s b)(s c), where s = a+b+c EXAMPLE 1 YOUR TURN 1 Find the area of the triangle; Find the area of the triangle b = 7, c = 8, A = 80 (A) a = 7, b = 9, C = 55 C b = 7 a A 80 c = 8 B This is SAS case; Area of a triangle is 1 bc sin(a) = 1 (7)(8) sin(80 ) (B) A = 100, b = 8, c = 11

159 DR. YOU: 018 FALL 159 EXAMPLE YOUR TURN Find the area of the triangle: Find the area of the triangle; a = 7, b = 9, c = 5 (A) a = 8, b = 5, c = 10 C b = 9 a=7 A c = 5 B First, s = = 10.5 Area of a triangle is (B) a = 11, b = 13, c = YOUR TURN A farmer has a triangular field with sides 10 yards, 170 yards, and 0 yards. Find the area of the field in square yards. YOUR TURN The courtyard for a building is in the shape of a triangle. An included angle measure is 5, two adjacent sides of the angle are 14 meters,1 meters. Find the area of the triangle.

160 DR. YOU: 018 FALL 160 NOTE: (A Right triangle) For the duration of this class, we will label the sides and angles of triangles so that the angles are in CAPITAL letters and the sides are in lowercase letters, and each side will bear the same letter as B the angle opposite it. Notice in the figure to the right that side "a" is opposite angle "A" and side "b" is opposite angle "B". And, as with c a any right triangle, a + b = c, and since the sum of all angles must equal 180, and the right angle is 90, then the other two angles must add up to A b C 90 (A + B = 90 ). EXAMPLE 3 YOUR TURN 3 Find the values of A, B, C, a, b and c in the figure. Find the values of A, B, C, a, b and c in the figure. 1) Pythagorean Theorem: a + b = c a = 10 5 = 75 = 5 3 ) cos A = 5 10 A = cos 1 ( 5 10 ) = 60 3) B = A = 30

161 DR. YOU: 018 FALL 161 EXAMPLE 4 EXAMPLE 5 Find the values of A, B, C, a, b and c in the figure. Use the given information to solve the right triangle. EXAMPLE 6 EXAMPLE 7 Use the given information to solve the right triangle. Find the values of A, B, C, a, b and c in the figure. b = 4, B = 10, C = 90 a =, b = 8, C = 90

162 DR. YOU: 018 FALL 16 APPLICATIONS IN NAVIGATION AND SURVEYING In navigation and surveying, the "direction" or "bearing" is the acute angle between any point and the north-south line. The angles are labeled as North-or-South (degree measurement) East-or-West N45 W N30 E S60 W S40 E EXAMPLE 8 A ship leaves the port of Miami with a bearing of S80 E and a speed of 15 knots. After 1 hour, the ship turns 90 toward the south. After more hours, maintaining the same speed, what is the bearing to the ship from port?

163 DR. YOU: 018 FALL 163 HOMEWORK 13 NAME As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to this coversheet. 1. Find the area of each triangle. Round answers to two decimal places. (A) a = 5, b = 8, c = 9 (B) a = 6, b = 4, C = 60 (C) a = 4, b = 18, c = 16 (D) A = 35, b = 8, c = 1. Find angles A, B and side a in a triangle if c = 5 and b = 14. Round to the nearest degree (C is the right angle). 3. Find angle A and sides a, b in a triangle if B = 35, c = 30. Round to the nearest degree (C is the right angle). 4. From a certain port, a ship travels 80-miles east and then turns 90 and travels 65 miles to the south. Find the bearing and distance of the ship from port. Round to the nearest tenth. 5. The dimension of home plate at any major league baseball stadium are shown. Find the area of home plate. 9 in 9 in 8 in 8 in 16 in 6. (extra) Find the area of the shaded region enclosed in a semicircle of diameter 10 inches. The length of the chord PQ is 8 inches. Q 8 in P 10 in R

164 DR. YOU: 018 FALL 164 LECTURE 3-4 SINE LAW THE LAW OF SINES b C a sin A a sin B = b = sin C c A c B EXAMPLE 1 YOUR TURN 1 (SAS case) Solve triangle ABC; Solve triangle ABC; B B c 96 a c a A 4 b =1 C A 4 46 b =1 C Since we know two angles, we can find the measure of C. C = = 4 Since b = 1 and B = 96, Similarly a = c = sin(4 ) a = sin(96 ) 1 a sin(4 ) = 1 sin(96 ) 1 sin(4 ) sin(96 ) sin(4 ) c = sin(96 ) 1 c sin(4 ) = 1 sin(96 ) 1 sin(4 ) sin(96 ) 8.073

165 DR. YOU: 018 FALL 165 EXAMPLE YOUR TURN Solve triangle ABC; A = 46, B = 63, c = 56 inches Solve triangle ABC; A = 50, C = 33, b = 76 1) Find C C = = 71 ) Find a sin(46 ) a = sin(71 ) 56 a sin(46 ) = 56 sin(71 ) a = 56 sin(46 ) sin(71 ) 4.6 3) Find b b = sin(63 ) b = sin(71 ) 56 b sin(63 ) = 56 sin(71 ) 56 sin(63 ) sin(71 ) 5.77

166 DR. YOU: 018 FALL 166 EXAMPLE 3 YOUR TURN 3 (SSA case-one triangle) Solve triangle ABC a = 10, b = 9, A = 50 Solve triangle ABC a = 80, b = 65, A = 43 To find B. sin(b) 9 = sin(50 ) 10 sin(b) = 9 sin(50 ) 10 B = sin 1 (9 sin(50 ) ) Since 90 < sin 1 a < 90 and one angle in triangle can be bigger than 90, we have to consider one angle in II quadrant. First case Second case A B = A + B It is impossible since sum of all inside angle in a triangle is 180 C To find c, sin(85.41 ) c = c = sin(50 ) sin(85.41 ) sin(50 ) 13.01

167 DR. YOU: 018 FALL 167 EXAMPLE 4 YOUR TURN 4 (SSA-no triangle) Solve triangle ABC a = 50, b = 70, A = 75 Solve triangle ABC a = 1, b = 4, and A = 10 To find B. sin(b) 70 = sin(75 ) 50 sin(b) = 70 sin(75 ) Since the value of sine NEVER exceed 1, there is no angle B for which sin(b) = B = sin 1 (70 sin(75 ) ) = sin 1 (1.35) : error 50 There is no triangle with the given condition.

168 DR. YOU: 018 FALL 168 EXAMPLE 5 YOUR TURN 5 (SSA-two triangles) Solve triangle ABC a = 54, b = 6, A = 40 Solve triangle ABC a = 7, b = 8, and A = 60 To find B. sin(b) 6 = sin(40 ) 54 sin(b) = 6 sin(40 ) 54 B = sin 1 (6 sin(40 ) ) Since 90 < sin 1 a < 90 and one angle in triangle can be bigger than 90, we also consider one angle in II quadrant. First case Second case A B = A + B C When B = 47.56, C = 9.44 case, sin(9.44 ) c = c = sin(40 ) sin(9.44 ) sin(40 ) When B = 13.44, C = 7.56 case, sin(7.56 ) c = c = sin(40 ) sin(7.56 ) sin(40 ) 11.05

169 DR. YOU: 018 FALL 169 EXAMPLE 6 YOUR TURN 6 (SSA-one triangle) Solve triangle ABC a =, b = 14, A = 0 Solve triangle ABC a = 8, b = 6, A = 15 To find B. sin(b) 14 = sin(0 ) sin(b) = 14 sin(0 ) B = sin 1 (14 sin(0 ) ) 1.57 Since 90 < sin 1 a < 90 and one angle in triangle can be bigger than 90, we also consider one angle in II quadrant. First case Second case A 0 0 B = A + B C : impossible since measure of angle is positive. When B = 1.57, C = case, sin(14.43 ) c = c = sin(0 ) sin(14.43 ) sin(0 ) 39.

170 DR. YOU: 018 FALL 170 YOUR TURN Solve triangle ABC b = 6, c = 8, and B = 35 YOUR TURN Solve triangle ABC a = 10, b = 4, A = 50

171 DR. YOU: 018 FALL 171 EXAMPLE 7 YOUR TURN 7 To find the distance across a canyon from point A to point C, a surveying team locates points B and C on one side of the canyon and a point A on the other side of the canyon. The distance between B and C is 100 feet. The angle ABC is 65 and the angle ACB is 80. Find the distance across the canyon from point C to point A. To find the distance across a canyon from point A to point C, a surveying team locates points B and C on one side of the canyon and a point A on the other side of the canyon. The distance between B and C is 150 feet. The angle ABC is 50 and the angle ACB is 110. Find the distance across the canyon from point C to point A. 1) Draw the figure A x B C ) Use sine law to find x (we know two angles) A = = 35 sin(35 ) 100 = sin(65 ) x 100 sin(35 ) = x sin(65 ) x = 100 sin(65 ) sin(35 ) ft

172 DR. YOU: 018 FALL 17 EXAMPLE 8 YOUR TURN 8 A state trooper is hidden 30 feet from a highway. One A state trooper is hidden 50 feet from a highway. One second after a truck passes, the angle θ between the second after a truck passes, the angle θ between the highway and the line of observation from the patrol car to highway and the line of observation from the patrol car to the truck is measured. If the angle measures 13, how fast the truck is measured. If the angle measures 15, how fast is the truck traveling? Express the answer in feet per is the truck traveling? Express the answer in feet per second and in miles per hour. second and in miles per hour. B C: 13 Truck 30 ft A: state trooper B C: 15 Truck 50 ft A: state trooper In the right triangle ABC, we find the distance BC. Since AB is the opposite side of C = 13 and BC is the adjacent side of C. tan(13 ) = AB BC = 30 BC BC tan(13 ) = 30 BC = 30 tan(13 ) Speed = distance Time = ft 1 sec ft 1 mile 3600 sec 1 sec 580 ft 1 hr = mi/hr Speed is ft/sec or mi/hr

173 DR. YOU: 018 FALL 173 EXAMPLE 9 YOUR TURN 9 A pole tilts toward the sun at an 8 angle from the vertical, Because of prevailing winds, a tree grew so that it was and it casts a -foot shadow. The angle of elevation from leaning 4 from the vertical. At a point 35 meters from the the tip of the shadow to the top of the pole is 43. How tall tree, the angle of elevation to the top of the tree is 3 (see is the pole? figure). Find the height h of the tree. 1) First, draw the figure. ) Find the angle of B and C B = = 98 C = = 39 3) Find the length of a by using sine law a sin(a) = c sin(c) a sin(43 ) = sin(39 ) a = sin(43 ) sin(39 ) ft Therefore, the length of the pole is ft

174 DR. YOU: 018 FALL 174 EXAMPLE 10 YOUR TURN 10 Find the distance CD, x. Find the value of x in the figure x ) Find angle B ) Find angle C 3) Find AC B = = 15 C = = 15 AC = sin(15 ) AC 100 sin(15 ) sin(15 ) = sin(15 ) ) Use the right triangle ACD to find x. sin(30 ) = x AC x sin(30 ) = x = sin(30 ) 158.5

175 DR. YOU: 018 FALL 175 EXAMPLE 11 YOUR TURN 11 Find the current height of the pyramid, using the information given in the figure. To find the height of a proposed ski lift, a surveyor measure angle DPQ to be 5 and the walk of a 1000-ft distance to R and measure angle PRQ to be 15. What is h the height of this ski lift? ft 100 ft 1) First, find x in the below figure. x 15 sin(130 ) x x = ft 130 = sin(15 ) sin(130 ) sin(15 ) ) Find the height in the below figure x h 35 sin(35 ) = h x h = x sin(35 ) = 50 sin(130 ) sin(35 ) sin(15 ) ft

176 DR. YOU: 018 FALL 176 EXAMPLE 1 YOUR TURN 1 A blimp, suspended in the air at a height of 400 ft., lies directly over a line from a sports stadium to a planetarium. If the angle of depression from the blimp to the stadium is 39 and from the blimp to the planetarium is 6, find the distance between the sports stadium and the planetarium. A balloonist is directly above a straight road that joins two villages. She is suspended in the air at a height of 500 m directly from the ground. She finds that the town closer to her is at an angle of depression of 35, and the farther town is at an angle of depression of 30. Find the distance blimp between two towns ft 6 Balloon m sport stadium Planetarium A B C A blimp lies directly above the ground point A. First, find the distance between sport stadium and A. 39 B 400 ft S A P sin(39 ) 400 SB = = sin(51 ) SB 400 sin(51 ) sin(39 ) Second, find the distance between the planetarium and A. 39 B 400 ft S A P sin(6 ) 400 PB = = sin(64 ) PB 400 sin(64 ) sin(6 ) The distance between the sports stadium and the planetarium is 400 sin(51 ) 400 sin(64 ) ft sin(39 ) sin(6 )

177 DR. YOU: 018 FALL 177 EXAMPLE 13 YOUR TURN 13 An airplane is spotted by two observers at A and B who The highest bridge in the world is the bridge over the are 1700 ft. apart. As the airplane passes over the line Royal Gorge of the Arkansas River in Colorado. joining them, each observer takes a sighting of the angle Sightings to the same point at water level directly under of elevation to the plane, as indicated in the figure. If A = the bridge are taken from each side of the 880-foot-long 50, and B = 0, how high is the airplane? bridge, as indicated in the figure. How high is the bridge? C airplane 880 ft A 50 h B h 1) Find angle C C = = 110 ) Find the distance AC sin(50 ) AC = sin(110 ) 1700 AC sin(50 ) = 1700 sin(110 ) AC = 1700 sin(50 ) sin(110 ) ) Use the right triangle to find h sin(50 ) = h h = sin(50 ) = C airplane h A B

178 DR. YOU: 018 FALL 178 HOMEWORK 14 NAME As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to this coversheet. 1. Find all sides and all angles of the triangle given below. Determine whether the given information results in one triangle, two triangles, or no triangle at all. (A) A = 59, C = 15, c = 10 yd (B) A = 100, c = 1 cm, a = 10 cm (C) A = 5, b = 8 ft, a = 7ft (D) A = 98, c = 11 cm, a = 19 cm. To find the distance across a canyon from point B to point C, a surveying team locates points A and B on one side of the canyon and point C on the other side of the canyon. The distance between A and B is 9 yards. Angle CAB measures 67, and angle CBA is 89. Find the distance across the canyon from point B to point C. Round to the nearest yard. 3. The famous Leaning Tower of Pisa was originally feet high. At a distance of 13 feet from the base of the tower, the angle of elevation to the top of the tower is found to be 60. Find the measure of angle RPQ indicated in the figure. Also, find the perpendicular distance from R to PQ. 4. An aircraft is spotted by two observers who are 1000 feet apart. As the airplane passes over the line joining them, each observer takes a sighting of the angle of elevation to the plane, as indicated in the figure. How high is the airplane? 5. A hiker is approaching a mountain. The top of the mountain is at an angle of elevation of 5. After the hiker crosses 800 ft of level ground directly towards the mountain, the angle of elevation becomes 9. Find the height of the mountain.

179 DR. YOU: 018 FALL 179 LECTURE 3-5 COSINE LAW THE LAW OF COSINES b C a c = a + b ab cos(c) ; cos(c) = a + b c ab b = a + c ac cos(b) ; cos(b) = a + c b ac A c B a = b + c bc cos(a) ; cos(a) = b + c a bc EXAMPLE 1 YOUR TURN 1 (SAS case) Solve triangle ABC; Solve triangle ABC; a = 7, b = 3, and C = 60 a =, b = 3, C = 60 1) c = a + b ab cos(c) = = ) a = b + c bc cos(a) 49 = cos(a) = 3 37 cos(a) = cos(a) 6 37 A = cos ( 6 37 ) ) B = = 5.85

180 DR. YOU: 018 FALL 180 EXAMPLE YOUR TURN (SSS case) Solve triangle ABC; Solve triangle ABC; a = 7, b = 3, and c = 6 a = 0, b = 5, c = 18 Since a is biggest, find the angle A 1) a = b + c bc cos(a) 49 = cos(a) = 3 6 cos(a) 4 = 36 cos(a) 4 = cos A 36 A = cos 1 ( 4 36 ) ) b = a + c ac cos(b) 9 = cos(b) = 7 6 cos(b) 76 = 84 cos(b) cos(b) = B = cos 1 ( ) ) C = = 58.41

181 DR. YOU: 018 FALL 181 EXAMPLE 3 YOUR TURN 3 Solve triangle ABC; Solve triangle ABC; a = 1, b = 9, and c = a = 7, b = 9, and c = 0 Since c is biggest, find the angle A c = a + b ab cos(c) = (1)(9) cos(c) 59 = 16 cos(c) cos(c) = 59 < 1 which is impossible 16 There is no triangle which satisfies the conditions. YOUR TURN Solve triangle ABC; b = 15, c = 1, and A = 4 YOUR TURN Solve triangle ABC; b = 5, c = 7, A = 55

182 DR. YOU: 018 FALL 18 YOUR TURN Solve triangle ABC; a = 1, b = 15, c = 0 YOUR TURN Solve triangle ABC; a = 7, b = 5, c = 10

183 DR. YOU: 018 FALL 183 EXAMPLE 4 YOUR TURN 4 Find the distance across the lake from A to C, to the nearest yard, using the measurement shown in the figure. To approximate the length of a marsh, a surveyor walks 50 meters from point A to point B, then turns 75 and A C walks 0 meters to point C (see figure). Approximate the length AC of the marsh. 140 yd yd B We want to find b = AC b = (140)(160) cos(80 ) b = (140)(160) cos(80 ) yd

184 DR. YOU: 018 FALL 184 EXAMPLE 5 YOUR TURN 5 A triangular parcel of land has 115 meters of frontage, and the other boundaries have lengths of 76 meters and 9 meters. What angles does the frontage make with the two A triangular parcel of ground has sides of lengths 75 feet, 650 feet, and 575 feet. Find the measure of the largest angle. other boundaries? Let θ be the angle between two other boundaries. 115 = (76)(9) cos θ = (76)(9) cos θ = cos θ (76)(9) 1015 = cos θ θ = cos 1 ( ) 85.84

185 DR. YOU: 018 FALL 185 EXAMPLE 6 YOUR TURN 6 An airplane flies due north from Ft. Myers to Sarasota, a A cruise ship maintains a speed of 10 knots (nautical distance of 150 miles, and then turns through an angle of miles per hour) sailing from San Juan to Barbados, a and flies to Orlando, a distance of 100 miles. See the 600- nautical mile distance. To avoid a tropical storm, the figure. captain heads out of San Juan at a direction of 1 off a direct heading to Barbados. The captain maintains the 10- knot speed for 5 hours, after which time the path to Barbados becomes clear of storms. Barbados 600 miles Ship San Juan (A) How far is it directly from Ft. Myers to Orlando? (B) What bearing should the pilot use to fly directly from Ft. Myers to Orlando? First, find the measure of angle Orlando-Sarasota- Ft.Myers: 130 (A) What angle should be the captain turn to head directly to Barbados? (B) If one the turn is made, how long will it be before the ship reaches Barbados if the same 10 knot speed is maintained? Use the cosine law to find the distance D from Ft. Myers to Orlando D = (150)(100) cos(130 ) D = Use the sine law to find the measure of angle Sarasota- Ft.Myers-Orlando, θ. sin θ 100 = sin(130 ) 7.56 sin θ = 100 sin(130 ) 7.56 θ = sin sin(130 ) ( )

186 DR. YOU: 018 FALL 186 EXAMPLE 7 YOUR TURN 7 A softball diamond is a square, 60 ft on a side, with home Suppose a certain baseball diamond is a square 75 ft. on a plate and the three bases as vertices. The pitcher s position side. The pitching rubber is located 49.5 ft. from home is 46 ft form home plate. Find the distance from the plate on a line joining home plate and second base. pitcher s position to each of the bases. (A) How far is it from the pitching rubber to first base? (B) How far is it from the pitching rubber to second base? First, draw the figure. (C) If a pitcher faces home plate, through what angel nd base does he need to turn to face first base? 3rd base 1st base pitcher Home home plate The segment from nd base to home is diagonal. By Pythagorean theorem. the diagonal = = ft So, the distance between pitcher and nd base is = ft Let x be the distance between pitcher and 1 st base. nd base 3rd base x Home 1st base By cosine Law, x = (46)(60) cos(45 ) x = (46)(60) cos(45 ) x ft The distance between pitcher and 3 rd base is x = 43.56

187 DR. YOU: 018 FALL 187 HOMEWORK 15 NAME As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to this coversheet. 1. Solve the triangle. (A) a = 6, b = 4, C = 60 (B) a = 3, c =, B = 110 (C) a = 1, b = 40, c = 3 (D) a = 1, b = 13, c = 5. The angle at one corner of a triangular plot of ground is 70, and the sides that meet at this corner are 175 feet and 150 feet long. Approximate the length of the third side. (Round your answer to the nearest whole number.) 3. The height of a radio tower is 700 feet, and the ground on one side of the tower slopes upward at an angle of 10 (see the figure). (A) How long should a guy wire be if it is to connect to the top of the tower and be secured at a point on the sloped side 10 feet from the base of the tower? (B) How long should a second guy wire be if it is to connect to the middle of the tower and be secured at a point 10 feet from the base on the flat side? 700 ft 10 ft 10 ft 10

188 DR. YOU: 018 FALL 188 REVIEW PROBLEMS FOR EXAM 3 Click the blue words to see video: 1. Evaluate the value of trigonometry by using the sum/difference formulas : Problem 1, 3 and 6. Find the value of cosine by using double angle formula: Problem 4 3. Find the value of trigonometry by using half angle formula: Problem 5 4. Evaluate the value of trigonometry involving inverse trig: Problem 7 5. Find the triangle area: Problem 8 and 9 6. Law of sines: Problem 10, 11, and 1 7. Law of cosines: Problem Application 1: Problem Application : Problem Write the following in a single trigonometry function: sin(3x) cos(5x) + cos(3x) sin(5x). Express sin(8x) + sin(6x) by using sum to product. 3. Find the exact value the following expression by using the sum/difference formulas. (A) sin(105 ) 4. Find the exact value of cos(θ) if sin(θ) = 1 and π < x < π 3 (B) cos ( 13π 1 ) 5. Find the exact value of each expression by using the half angle formula (A) sin(15 ) 6. Find the exact value of sin(a + b) if sin a = 4 5, π (B) tan(.5 ) < a < π and cos(b) = 1 13, π < b < 0 7. Find the exact value of sin [sin 1 ( 3 5 ) cos 1 ( 5 13 )] 8. Find the area of the triangle such that A = 75, b = 9 cm, c = 1 cm 9. Find the area of the triangle such that a = 7 cm, b = 8 cm, c = 9 cm 10. Solve the following triangle (find remaining sides and angles) if possible: A = 70, B = 60, c = 4

189 DR. YOU: 018 FALL Solve the following triangle if possible: A = 100, c = 1, a = Solve the following triangle if possible: A = 70, c = 8, a = Solve the following triangle if possible: A = 40, c = 3, a = Solve the following triangle if possible: a = 10, c = 0, B = To find the distance across a canyon from point B to point C, a surveying team locates points A and B on one side of the canyon and point C on the other side of the canyon. The distance between A and B is 9 yards. Angle CAB measures 67, and angle CBA is 89. Find the distance across the canyon from point B to point C. Round to the nearest yard. 16. As shown in the figure below, a cable car carries passengers from a point A, which is 1. miles from a point B at the base of a mountain, to a point P at the top of the mountain. The angles of elevation of P from A and B are A = 1 and B = 65, respectively. Find the height of the mountain. (Round your answers to one decimal place.) P A miles B h

190 DR. YOU: 018 FALL 190 SOLUTIONS 1. By using sum of sine:. sin(3x) cos(5x) + cos(3x) sin(5x) = sin(3x + 5x) = sin(8x) 8x + 6x 8x 6x sin(8x) + sin(6x) = sin ( ) cos ( ) = sin(7x) cos(x) 3. (A) sin(105 ) = sin( ) = sin(60 ) cos(45 ) + cos(60 ) sin(45 ) = = (B) cos ( 13π ) = cos(195 ) = cos( ) 1 = cos(150 ) cos(45 ) sin(150 ) sin(45 ) = ( 3 ) 1 4. = 6 4 cos(θ) = 1 (sin θ) = 1 ( 1 3 ) = (A) sin(15 ) = sin ( 30 ) = + 1 cos(30 ) since 15 lies in I quadrant and sin(15 ) > = + = (B) tan(.5 ) = tan ( 45 ) = 1 cos( 45 ) sin( 45 ) = 1 (1 3 ) () = 3 4 = 3 = (1 ) = = + = ( + ) = = 1 6. a : II x = 3 y = 4 r = 5 b: IV x = 1 y = 5 r = 13 sin(a + b) = sin a cos b + cos a sin b = 4 5 (1) + ( 3 ) ( 5 ) =

191 DR. YOU: 018 FALL a = sin 1 ( 3 5 ) : I x = 4 y = 3 r = 5 b = cos 1 ( 5 13 ): II x = 5 y = 1 r = 13 sin(a b) = sin a cos b cos a sin b = 3 5 ( 5 13 ) (4 5 ) (1 13 ) = (9)(1) sin(75 ) Let s = = 1 1(1 7)(1 8)(1 9) Sine Law (two angles) 1) C = = 50. ) Since we know C = 50 and c = 4, sin(c) c sin(c) c = sin(b) b = sin(a) a sin(50 ) 4 sin(50 ) 4 = sin(60 ) b = sin(70 ) a b sin(50 ) = 4 sin(60 ) b = 4 sin(60 ) sin(50 ) 4.5 a sin(50 ) = 4 sin(70 ) b = 4 sin(70 ) sin(50 ) Sine Law (a pair A = 100 and a = 10) 1) sin(a) a = sin(c) c sin(100 ) 10 = sin(c) 1 10 sin(c) = 1 sin(100 ) sin(c) = 1 sin(100 ) 10 It is impossible. There are no such triangles. 1. Sine Law (a pair A = 70 and a = 10) =.07 but 1 sin(c) 1 1) sin(a) a = sin(c) c sin(70 ) 10 = sin(c) 8 C = sin 1 ( 8 sin(70 ) ) sin(c) = 8 sin(70 ) sin(c) = 8 sin(70 ) 10 ) Check whether it is one triangle case or two triangles case First case Second case A C = B 61.6 Not a triangle It is only one triangle case. 3) To find b sin(a) a = sin(b) b sin(70 ) 10 = sin(61.6 ) b b = 10 sin(61.6 ) sin(70 ) 9.33 One triangle: C = 48.74, B = 61.6, b = 9.33

192 DR. YOU: 018 FALL Sine Law (a pair A = 40 and a = ) 1) sin(a) a = sin(c) c sin(40 ) = sin(c) 3 C = sin 1 ( 3 sin(40 ) ) 74.6 ) Check whether it is one triangle case or two triangles case It is two triangles case. 3) To find b Case 1: sin(a) a Case 1: sin(a) a = sin(b) b = sin(b) b First case sin(c) = 3 sin(40 ) sin(c) = 3 sin(40 ) Second case A C = B sin(40 ) sin(40 ) = sin(65.4 ) b = sin(34.6 ) b b = sin(65.4 ) sin(40 ) b = sin(34.6 ) sin(40 ) Two triangles: Case1: C = 74.6, B = 65.4, b =.83 Case1: C = 105.4, B = 34.6, b = Cosine Law (no pair) a = 10, c = 0, B = 110 1) b = a + c ac cos(b) b = (10)(0) cos(110 ) b = 5.4 ) a = b + c bc cos(a) cos(a) = b +c a bc A = cos 1 ( (5.4) + (0) 10 ) 1.85 (5.4)(0) 3) C = = We know that A = 67, B = 89, c = 9 yards. We want to find a Then C = = 4 sin(a) a = sin(c) c sin(67 ) a = sin(4 ) 9 a = 9 sin(67 ) sin(4 ) = (5.4) +(0) 10 (5.4)(0) 08.9 yards 16. In ABP, A = 1, B = = 115, P = = 44, p = 1. sin(p) p = sin(b) b sin(44 ) 1. = sin(115 ) b In the right triangle, A = 1, AP = (hypothenuse) To find h (opposite side) sin(1 ) = h b = 1. sin(115 ) sin(44 ) h = sin(1 ) miles =

193 DR. YOU: 018 FALL 193 CHAPTER 4. POLAR COORDINATES LECTURE 4-1 POLAR COORDINATES NOTE: NOTE: We know that tan 1 ( y ) if θ is in I quadrant x x = r cos θ, y = r sin θ, and r = x + y, θ = tan 1 ( y ) if θ is in II, III quadrant x Standard form or rectangular form { tan 1 ( y ) if θ is in IV quadrant x Polar form magnitude Coordinates (x, y) (r, θ) r = x + y Complex number x + yi where i = 1 r cos θ + ir sin θ = r(cos θ + i sin θ) x + yi = x + y Vectors x, y = xi + yj r cos θ i + r sin θ j = r(cos θ i + sin θ j) xi + yj = x + y y r θ (r, θ) polar coordinate x Polar coordinates case: the expression is not unique. (r, θ) = (r, θ + nπ) = ( r, θ + π + nπ) for any integer n EXAMPLE 1 YOUR TURN 1 Plot the point with the polar coordinate (, 10 ) Plot the point with the polar coordinate (5,10 ) 10 (, 10 )

194 DR. YOU: 018 FALL 194 EXAMPLE YOUR TURN Plot the point with the polar coordinate (, 150 ) Plot the point with the polar coordinate (5, 40 ) (, 150 ) = (, ) = (,10 ) (, 150 ) 150 EXAMPLE 3 YOUR TURN 3 Plot the point with the polar coordinate (, 30 ) Plot the point with the polar coordinate ( 5, 60 ) (, 30 )= (, ) = (,10 ) 30 (, 30 ) EXAMPLE 4 YOUR TURN 4 Plot the point with the polar coordinate (, 390 ) Plot the point with the polar coordinate ( 5,300 ) (, 390 )= (, ) = (,570 ) = (,10 ) (, 390 ) 390

195 DR. YOU: 018 FALL 195 EXAMPLE 5 YOUR TURN 5 Rewrite the polar coordinate (, (r, θ) such that (A) r > 0, π θ < 0 5π 6 ) as a polar coordinate Rewrite the polar coordinate (8, 70 ) as a polar coordinate (r, θ)such that (A) r > 0, π θ < 0 r = > 0 is same The coterminal angle θ in π θ < 0 is θ = 5π 6 π = 7π 6 (, 5π 6 ) = (, 7π 6 ) (B) r > 0, π θ < 4π (B) r > 0, π θ < 4π (, 5π 6 ) = (, 5π 6 + π) = (, 19π 6 ) (C) r < 0, 0 θ < π (C) r < 0, 0 θ < π (, 5π 6 ) = (, 5π 6 + π) = (, 11π 6 )

196 DR. YOU: 018 FALL 196 EXAMPLE 6 YOUR TURN 6 Rewrite the polar coordinate ( 3, coordinate (r, θ)such that (A) r < 0, π θ < 0 π 3 ) as a polar Rewrite the polar coordinate ( 4, 60 ) as a polar coordinate (r, θ)such that (A) r < 0, 0 θ < π The coterminal angle of π 3 is π 3 π = 4π 3 So, ( 3, π 3 ) = ( 3, 4π 3 ) (B) r > 0, 0 θ < π 1) First, find the equivalent form but r > 0 (B) r > 0, 0 θ < π ( 3, π 3 ) = (3, π 3 + π) = (3, 5π 3 ) ) Check that 5π 3 lies on 0 θ < π : Ture Then ( 3, π 3 ) = (3, 5π 3 ) (C) r > 0, π θ < 4π 1) First, find the equivalent form but r > 0 (C) r > 0, π θ < 4π ( 3, π 3 ) = (3, π 3 + π) = (3, 5π 3 ) ) Find coterminal angle of 5π 3 on π θ < 4π Then ( 3, π 3 5π 3 ) = (3, 11π 3 ) + π = 11π 3

197 DR. YOU: 018 FALL 197 EXAMPLE 7 YOUR TURN 7 Find the rectangular coordinates of the given polar coordinates (A) (6, π 6 ) Find the rectangular coordinates of the polar coordinates (A) (3, 5π 3 ) Find x, y where r = 6 and θ = π 6 x = r cos θ = 6 cos ( π 6 ) = 6 3 = 3 3 y = r sin θ = 6 sin ( π 6 ) = 6 1 = 3 Then, the rectangular coordinate of this point is (x, y) = (3 3, 3) (B) ( 5, 11π 6 ) (B) ( 4, π 4 ) Find x, y where r = 4 and θ = π 4 x = r cos θ = 4 cos ( π ) = 4 4 = y = r sin θ = 4 sin ( π ) = 4 4 = Then, the rectangular coordinate of this point is (x, y) = (, ) (C) (4, 3π 4 )

198 DR. YOU: 018 FALL 198 EXAMPLE 8 YOUR TURN 8 Find the polar coordinates of the following rectangular Find the polar coordinates of the following rectangular coordinates. coordinates. (A) (1, 3) First, find r (A) ( 3, ) r = (1) + ( 3) = 4 = Find the angle whose terminal point is (1, 3) It is in IV quadrant: θ = tan 1 ( 3 ) = Then, its polar coordinate is (r, θ) = (, 300 ) (B) ( 4, 4) (B) ( 3,3) First, find r r = ( 3) + (3) = 18 = 3 Find the angle whose terminal point is ( 3,3) It is in II quadrant: θ = tan 1 ( 3 ) = Then, its polar coordinate is (r, θ) = (3, 135 ) (C) (,3)

199 DR. YOU: 018 FALL 199 EXAMPLE 10 YOUR TURN 10 Convert the following equation to a polar equation. Convert the following equation to a polar equation. x + y = 8x (A) 4xy = 9 We know that x = r cos θ y = r sin θ x + y = 8x Divide both sides by x + y = 4x Plug in x = r cos θ, y = r sin θ (r cos θ) + (r sin θ) = 4r cos θ r cos θ + r sin θ = 4r cos θ r {cos θ + sin θ} = 4r cos θ Since cos θ + sin θ = 1 (B) 10x + 10y = 80y r = 4r cos θ r = 4 cos θ Divide both sides by r since r > 0

200 DR. YOU: 018 FALL 00 EXAMPLE 11 YOUR TURN 11 Convert the following equation to a rectangular equation. (A) r = 6 cos θ Convert the following equation to a rectangular equation. (A) r = 8 cos θ Since r = x + y, cos θ = x r = x x +y x x + y = 6 x + y ( x + y ) = 6x x + y = 6x x 6x + y = 0 x 6x y = 9 (x 3) + y = 3 This is a circle with center (3,0) and radius 3 (B) r = 4 sin θ (B) r = 6 sin θ Since r = x + y, sin θ = y r = y x +y x + y y = 4 x + y ( x + y ) = 4y x + y = 4y x + y 4y = 0 x + y 4x + 4 = 4 x + (y ) = This is a circle with center (0,) and radius

201 DR. YOU: 018 FALL 01 EXAMPLE 1 YOUR TURN 1 Convert the following equation to a rectangular equation. (A) r sin θ = 3 Since r = x + y, sin θ = y r = x + y y x + y = 3 y = 3 (horizontal line) y x +y, Convert the following equation to a rectangular equation. (A) r cos θ = 4 (B) r = 4 (B) r = Since r = x + y x + y = 4 take square in both sides x + y = 4 (circle with center origin and radius 4) (C) θ = 45 It is a line with slope tan(45 ) = 1 and containing the origin; y = x (C) θ = π 6

202 DR. YOU: 018 FALL 0 HOMEWORK 16 NAME As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to this coversheet. 1. Find the equivalent polar coordinate (r, θ) of (, 140 ) which satisfies the conditions. (A) r > 0, π θ < 0 (B) r < 0, 0 θ < π (C) r > 0, π θ < 4π. Find the equivalent rectangular coordinates of the following polar coordinates. (A) (5, 300 ) (B) (, 3π 4 ) (C) ( 4, π 3 ) 3. Find the equivalent polar coordinates of the following rectangular coordinates. (A) ( 8,8) (B) (0, 4) (C) ( 3, ) 4. Let x and y represent rectangular coordinates. Write each rectangular equation as the polar equation. x + y = 4x 5. Let r and θ represent polar coordinates. Write each polar equation as the rectangular equation. Then identify and graph the equation. (A) r = cos θ (B) r = 5 sin θ

203 DR. YOU: 018 FALL 03 LECTURE 4- COMPLEX NUMBERS NOTE: COMPLEX PLANE PRODUCTS AND QUOTIENTS OF COMPLEX NUMBERS IN POLAR FORM: Let z 1 = r 1 (cos(θ 1 ) + i sin(θ 1 )) and z = r (cos(θ ) + i sin(θ )) 1) z 1 z = r 1 r (cos(θ 1 + θ ) + i sin(θ 1 + θ )) ) z 1 z = r 1 r (cos(θ 1 θ ) + i sin(θ 1 θ )) 3) [r(cos θ + i sin θ)] n = r n (cos(nθ) + i sin(nθ)) n 4) The n-th roots of z = r(cos θ + i sin θ) are r (cos ( θ + πk ) + i sin n n (θ + πk ) ), k = 0, 1,, n 1 n n HOW TO CONVERT FORMS?: Click to see the video HOW TO FIND THE PRODUCT/DIVISION OF COMPLEX NUMBERS?: 1) Case 1 ) Case 3) Case 3 EXAMPLE 1 YOUR TURN 1 Plot the points in the complex plane and Plot the points in the complex plane 3 i 4 + i

204 DR. YOU: 018 FALL 04 EXAMPLE YOUR TURN Find the absolute value of the complex number 4 + 3i Find the absolute value of the complex number (A) 3 3i 4 + 3i = ( 4) + (3) = = 5 = 5 (B) i EXAMPLE 3 YOUR TURN 3 Find the conjugate of the complex number 3 i Find the conjugate of the complex number (A) 4 5i The conjugate of 3 i is 3 + i (B) 3 + 8i (C) 6i

205 DR. YOU: 018 FALL 05 EXAMPLE 4 YOUR TURN 4 Find the standard form of the polar form. Find the standard form of the polar form. (A) (cos(60 ) + i sin(60 )) (A) (cos(10 ) + i sin(10 )) Since r =, θ = 60, x = r cos θ = cos(60 ) = 1 = 1 y = r sin θ = sin(60 ) = 3 = (cos(60 ) + i sin(60 )) = 1 + i (B) 6(cos(10 ) + i sin(10 )) (B) 8(cos(5 ) + i sin(5 )) Since r = 6, θ = 10, x = 6 cos(10 ) = 6 3 = 3 3 y = sin(60 ) = 6 1 = 3 6(cos(10 ) + i sin(10 )) = 3 3 3i (C) 4 (cos ( 5π 6 ) + i sin (5π 6 )) (C) 6 (cos ( 7π ) + i sin (7π )) 1 1 r = 3, θ = π 5 x = 3 cos ( π 5 ).47 y = 3 sin ( π 5 ) (cos ( π 5 ) + i sin (π )) = i 5

206 DR. YOU: 018 FALL 06 EXAMPLE 5 YOUR TURN 5 Find the polar form of the standard form. Find the polar form of the standard form. i (A) 4 + 4i We know that x =, y = 1) r = + = ) Find θ such that the terminal point is (, ) It is in III quadrant: θ = tan 1 ( ) = 5 So, i = (cos(5 ) + i sin(5 )) (B) 3 3 3i (C) 5i

207 DR. YOU: 018 FALL 07 EXAMPLE 6 YOUR TURN 6 Find z 1 z and z 1 z z = 8(cos(0 ) + i sin(0 )). if z 1 = (cos(50 ) + i sin(50 )) and Find z 1 z and z 1 z z = 8(cos(5 ) + i sin(5 )). if z 1 = (cos(85 ) + i sin(85 )) and We know that r 1 =, θ 1 = 50 in z 1 r = 8, θ = 0 in z To find z 1 z r = r 1 r = ()(8) = 16, θ = θ 1 + θ = = 70 z 1 z = 16(cos(70 ) + i sin(70 )) To find z 1 z r = r 1 r = 8 = 1 4, θ = θ 1 θ = 50 0 = 30 z 1 = 1 (cos(30 ) + i sin(30 )) z 4

208 DR. YOU: 018 FALL 08 EXAMPLE 9 YOUR TURN 9 Find z 1 z and z 1 z z = 3(cos(80 ) + i sin(80 )). if z 1 = 6(cos(110 ) + i sin(110 )) and Find z 1 z and z 1 z z = 8(cos(50 ) + i sin(50 )) if z 1 = 4(cos(170 ) + i sin(170 )) and (we usually write it as polar form and 0 θ < π) We know that r 1 = 6, θ 1 = 110 in z 1 r = 3, θ = 80 in z To find z 1 z r = r 1 r = (6)(3) = 18, θ = θ 1 + θ = = 390 (390 and 30 are coterminal) z 1 z = 18(cos(390 ) + i sin(390 )) = 18(cos(30 ) + i sin(30 )) To find z 1 z r = r 1 r = 6 3 =, θ = θ 1 θ = = 170 ( 170 and 190 are coterminal) z 1 z = (cos( 170 ) + i sin( 170 )) = (cos(190 ) + i sin(190 ))

209 DR. YOU: 018 FALL 09 EXAMPLE 10 YOUR TURN 10 Find z 1 z and z 1 z z = 6 (cos ( 9π ) + i sin (9π )) if z 1 = 3 (cos ( 7π ) + i sin 8 (7π )) and 8 Find z 1 z and z 1 z z = 6 (cos ( 7π ) + i sin (7π )) if z 1 = 3 (cos ( 4π ) + i sin 5 (4π )) and 5 (we usually write it as polar form and 0 θ < π) We know that r 1 = 3, θ 1 = 7π 8 in z 1 r = 6, θ = 9π 16 in z To find z 1 z r = r 1 r = (6)(3) = 18, θ = θ 1 + θ = 7π + 9π = 3π z 1 z = 18 (cos ( 3π ) + i sin (3π )) To find z 1 z r = r 1 r = 3 6 = 1, θ = θ 1 θ = 7π 9π = 5π z 1 = 1 (cos (5π) + i sin (5π z ))

210 DR. YOU: 018 FALL 10 EXAMPLE 11 YOUR TURN 11 Find z 1 z and z 1 z (write it as r(cos θ + i sin θ)) z 1 = 7 7i and z = 3 + i Find z 1 z and z 1 z (write it as r(cos θ + i sin θ)) z 1 = 1 + 3i and z = 3 + 3i We know that r 1 = (7) + ( 7) = 98 = 7 θ 1 = 315 since its terminal point is (, ) r = ( 3) + (1) = θ = 30 since its terminal point is ( 3, 1 ) To find z 1 z r = r 1 r = (7 )() = 14, θ = θ 1 + θ = = 345 z 1 z = 14 (cos(345 ) + i sin(345 )) To find z 1 z r = r 1 r = 7, θ = θ 1 θ = = 85 z 1 = 7 (cos(85 ) + i sin(85 )) z

211 DR. YOU: 018 FALL 11 EXAMPLE 1 YOUR TURN 1 Find z 1 z and z 1 z (write it as r(cos θ + i sin θ)) z 1 = 3 i and z = 1 + i Find z 1 z and z 1 z (write it as r(cos θ + i sin θ)) z 1 = 6 6 3i and z = i We know that r 1 = ( 3) + ( ) = 16 = 4 θ 1 = 300 since its terminal point is ( 3, 1 ) r = (1) + (1) = θ = 45 since its terminal point is (, ) To find z 1 z r = r 1 r = (4)( ) = 4, θ = θ 1 + θ = = 345 z 1 z = 18(cos(345 ) + i sin(345 )) To find z 1 z r = r 1 r = 4 =, θ = θ 1 θ = = 55 z 1 z = (cos(55 ) + i sin(55 ))

212 DR. YOU: 018 FALL 1 EXAMPLE 13 YOUR TURN 13 Simplify the expression in the standard form x + yi. ( (cos ( 5π 4 ) + i sin (5π ))) Simplify the expression in the standard form x + yi. ( (cos ( π 18 ) + i sin ( π 18 ))) 6 Then, r = ( ) 4 = 4 and θ = 4 5π 16 = 5π 4 ( (cos ( 5π 4 ) + i sin (5π ))) = 4 (cos ( 5π 4 ) + i sin (5π 4 )) x = 64 cos ( 5π ) = 4 = 4 y = 64 sin ( 5π ) = 4 = 4 = i EXAMPLE 14 YOUR TURN 14 Simplify the expression in the standard form x + yi. Simplify the expression in the standard form x + yi. (4(cos(15 ) + i sin(15 ))) 3 ((cos(10 ) + i sin(10 ))) 3 Then r = (4) 3 = 64 and θ = 3 15 = 45 (4(cos(15 ) + i sin(15 ))) 3 = 64(cos(45 ) + i sin(45 )) x = 64 cos(45 ) = 64 y = 64 sin(45 ) = 64 = i = 3 = 3

213 DR. YOU: 018 FALL 13 EXAMPLE 15 YOUR TURN 15 Simplify the expression in the standard form x + yi. Simplify the expression in the standard form x + yi. ( 3 + i) 5 (1 + i) 5 In 3 + i, r 1 = ( 3) + (1) = θ 1 = tan 1 ( 1 ) = 30 since the terminal point 3 ( 3, 1) is in I quadrant Then ( 3 + i) 5 = ((cos(30 ) + i sin(30 ))) 5 r = () 5 = 3 and θ = 5 30 = 150 ( 3 + i) 5 = ((cos(30 ) + i sin(30 ))) 5 = 3(cos(150 ) + i sin(150 )) x = 3 cos(150 ) = 3 3 = 16 3 y = 3 sin(150 ) = 3 1 = 16 = i

214 DR. YOU: 018 FALL 14 EXAMPLE 16 YOUR TURN 16 Find all the complex roots. Leave your answer in polar form: the complex cube roots of 64 Find all the complex roots. Leave your answer in polar form: the complex fourth roots of 16 We find 64 in the polar form 1) Find polar form of = 64(cos(0) + i sin(0)) r = 64; the angle 0 at (1,0) in unit circle ) The solutions are 3 64 where k = 0,1,. (cos (0 + πk πk ) + i sin ( )) In other words, 4(cos(0) + i sin(0)) 4 (cos ( π 3 ) + i sin (π 3 )) 4 (cos ( 4π 3 ) + i sin (4π 3 ))

215 DR. YOU: 018 FALL 15 HOMEWORK 17 NAME As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus. Attach all work to this coversheet. 1. Write each complex number in rectangular form (A) (cos(10 ) + i sin(10 )) (B) 3(cos(10 ) + i sin(10 )) (C) 4 (cos ( 7π 4 ) + i sin (7π 4 )) (D) (cos (5π 6 ) + i sin (5π 6 )). Write each complex number in polar form (A) 8 + 8i (C) 10i (B) 4 3 4i 3. Find zw and z. Leave your answer in polar form. w (A) z = (cos(80 ) + i sin(80 )) ; w = 6(cos(00 ) + i sin(00 )) (B) z = 3 (cos ( 3π 8 ) + i sin (3π 8 )) ; w = 4 (cos (9π) + i sin (9π )) Write each expression in the standard form a + bi. (A) [(cos(40 ) + i sin(40 ))] 3 (B) [ 3 (cos ( 5π ) + i sin (5π ))]6 5. Find all the complex roots. Leave your answer in polar form The complex cube roots of 8

216 DR. YOU: 018 FALL 16 LECTURE 4-3 VECTORS ALGEBRA VECTOR: v = Head point P (x, y ) tail point Q (x 1, y 1 ) = QP = (x x 1 )i + (y y 1 )j MAGNITUDE OR NORM: Let v = xi + yj be a vector: the length of v, v = x + y = r i, j rectangular expression Polar expression v xi + yj v (cos θ i + sin θ j) x = r cos θ, y = r sin θ θ is direction angle of v UNIT VECTOR OF A VECTOR :The magnitude is 1 in the same direction of v v = v = cos θ i + sin θ j v EXAMPLE 1 Let v = PQ be a vector from initial point P to terminal point Q. Write v in terms of i and j. (A) P = (3,), Q = (5,6) (B) P = ( 4,), Q = ( 1,5) v = (x x 1 )i + (y y 1 )j = (5 3)i + (6 )j = i + 4j (C) P = (, 1), Q = (6, ) (D) P = (,5), Q = ( 7, )