5.1: Angles and Radian Measure Date: Pre-Calculus

Transcription

1 5.1: Angles and Radian Measure Date: Pre-Calculus *Use Section 5.1 (beginning on pg. 482) to complete the following Trigonometry: measurement of triangles An angle is formed by two rays that have a common endpoint. One ray is called side and the other the side. Draw and label the diagram: An angle is in standard position if: Its vertex is at the origin of a rectangular coordinate system Its initial side lies along the positive x-axis Draw and label both diagrams: Positive angles are generated by a counterclockwise rotation. Negative angles are generated by a clockwise rotation. If the terminal side of an angle lies in a specific quadrant (I, II, III, IV), we say that the angle lies in that quadrant. If the terminal side of an angle lies on the x-axis or the y-axis, the angle is called a quadrantal angle. Measuring Angles Using Degrees Think about a clock. One full rotation is 60 minutes and this is a rotation of 360. Fractional parts of degrees are measured in minutes and seconds One minute, written 1 is degree: 1 =. One second, written 1 is degree: 1 =. For example, = ( ) Many calculators have keys for changing an angle from degree-minute-second notation (D M S ) to a decimal form and vice versa.

2 Measuring Angles Using Radians: Use the circle at the right: 1. Create an angle with the vertex at the center of the circle (a central angle) 2. Label the sides of the angle, r, for the radius If the central angle intercepts an arc along the circle measuring r units, the measure of such an angle is 1 radian 3. Label the arc created by the central angle r 4. Label the central angle 1 radian *1 radian = when arc and radius are equal The radian measure of any central angle is the length of the intercepted arc divided by the circle s radius. Construct an angle β with measure 2 radians. Radian Measure: Consider an arc of length s on a circle of radius r. The measure of the central angle, θ ( theta ), that intercepts the arc is r θ = s r radians. θ r s Relationship between Degrees and Radians: Compare the number of degrees and the number of radians in one complete rotation. θ = s r = 2πr r = 2π So, 360 = 2π radians. Thus, 180 = π radians. Conversion between Degrees and Radians: Using the basic relationship that 180 = π radians, 1. To convert degrees to radians, multiply degrees by 2. To convert radian to degrees, multiply radians by Angles that are fractions of a complete rotation are usually expressed in radian measure as factional multiples of π rather than as decimal approximations. For example θ = π 2.

3 Ex. 2: Convert each angle in degrees to radians: a) 60 b) 270 c) 300 Ex. 3: Convert each angle in radians to degrees: a) π 4 radians b) 4π 3 radians c) 6 radians Drawing Angles in Standard Position: *Goal: Learn to THINK in RADIANS Ex. 4: Draw and label each angle in standard position. a) θ = π 4 b) α = 3π 4 c) β = 7π 4 d) γ = 13π 4 Terminal Side Radian Measure of Angle Degree Measure of Angle 1 12 revolution 1 8 revolution 1 6 revolution 1 4 revolution 1 3 revolution

4 1 2 revolution 2 3 revolution 3 4 revolution 7 8 revolution 1 revolution Coterminal Angles- two angles with the same and sides. Every angle has infinitely many coterminal angles. Increasing or decreasing the degree measure of an angle in standard position by an integer multiple of 360 results in a coterminal angle. Thus, an angle of θ is coterminal with angles of θ ± 360 k, where k is an integer. Increasing or degreasing the radian measure of an angle by an integer multiple of 2π results in a coterminal angle. Thus, an angle of θ radians is coterminal with angles of θ ± 2πk, where k is an integer. Ex. 5: Find a positive angle less than 360 that is coterminal with each of the following: a) a 400 angle b) a 135 angle Ex. 6: Find a positive angle less than 2π that is coterminal with each of the following: a) a 13π 5 angle b) a π 15 angle Ex. 7: Find a positive angle less than 360 or 2π that is coterminal with each of the following: a) an 855 angle b) a 17π 3 angle c) a 25π 6 angle

5 5.1 Continued: Angles and Radian Measure Date: Pre-Calculus The Length of a Circular Arc: Ex 1: The minute hand of a clock is 6 inches long and moves from 12 to 4 o clock. How far does the tip of the minute hand move? Express your answer in terms of π and then round to two decimal places. Ex 2: The figure shows a highway sign that warns of a railway crossing. The lines that form the cross pass through the circle s center and intersect at right angles. If the radius of the circle is 24 inches, find the length of each of the four arcs formed by the cross. Express your answer in terms of π and then round to two decimal places. Ex 3: A circle has a radius of 6 inches. Find the length of the arc intercepted by a central angle of 45. Express arc length in terms of π. Then round your answer to two decimal places.

6 Linear and Angular Speed: If a point is in motion on a circle of radius r through an angle of θ radians in time t, then its LINEAR SPEED is: and its ANGULAR SPEED is: Note: The linear speed is the product of the radius and the angular speed, where w is the angular speed in radians per unit of time. Ex 4: The angular speed of a point on Earth is π radian per hour. The Equator lies on a circle of radius 12 approximately 4000 miles. Find the linear velocity, in miles per hour, of a point on the Equator. Ex 5: A Ferris wheel has a radius of 25 feet. The wheel is rotating at two revolutions per minute. Find the linear speed, in feet per minute, of a seat on this Ferris wheel. Ex 6: A water wheel has a radius of 12 feet. The wheel is rotating at 20 revolutions per minute. Find the linear speed, in feet per minute, of the water. Ex. 9: Long before ipods that hold thousands of songs and play them with superb audio quality, individual songs were delivered on 75-rpm and 45-rpm circular records. A 45-rpm record has an angular speed of 45 revolutions per minute. Find the linear speed, in inches per minute, at the point where the needle is 1.5 inches from the record s center. Homework: MathXL 5.1 Homework

7 5.2: Right Triangle Trigonometry Date: Pre-Calculus Fundamental Identities: Trigonometry Identities: Quotient Identities: Pythagorean Theorem: Divide both sides by c 2 What do you observe? Pythagorean Theorem Identities:

8 Ex 1: Given that sinθ = 1 2 and θ is an acute angle, find the value of cosθ using a trig identity. Consider the relationship between the sin30 =, cos60 =. So, sinθ = cos( ) Any pair of trigonometric functions f and g for which f(θ) = g(90 θ) are called. Cofunction Identities: If θ is measured in radians, replace the 90 with π 2. The value of a trigonometric function of θ is equal to the cofunction of the complement of θ. Cofunctions of complementary angles are equal. Ex 2: Find a cofunction with the same value as the given expression: a) sin46 b) cot π 12 Using a calculator to evaluate trigonometric functions: Ex 3: Using a calculator to find the value to four decimal places: a) sin72.8 b) csc1.5

9 Applications: The angle formed by a horizontal line and the line of sight to an object that is above the horizontal line is called the of. The angle formed by a horizontal line and the line of sight to an object that is below the horizontal line is called the of. Ex. 4: A flagpole that is 14 meters tall casts a shadow 10 meters long. Find the angle of elevation to the sun to the nearest degree. Ex. 5: Find the exact value of each expression. Do not use a calculator. a) tan π sec π 6 b) 1 + sin sin 2 50 c) cos12 sin78 + cos78 sin12 Ex. 6: Express the exact value of the function as a single fraction. Do not use a calculator. If f(θ) = 2sinθ sin θ 2, find f (π 3 ).

10 Ex. 7: At a certain time of day, the angle of elevation of the sun is 40. To the nearest foot, find the height of a tree whose shadow is 35 feet long. Ex. 8: The Washington Monument is 555 feet high. If you are standing one quarter of a mile from the base of the monument and looking to the top, find the angle of elevation to the nearest degree. Homework: Math XL: 5.2 Homework

11 5.3: Trigonometric Functions of Any Angle Date: Pre-Calculus Trigonometric Functions of Any Angle: The point P = (x,y) is a point r units away from the origin on the terminal side of θ. A right triangle is formed by drawing a line segment from P = (x,y) to the x-axis. Note: y is the length of the side opposite θ and x is the length of the side adjacent to θ. Definitions of Trigonometric Functions of Any Angle: Let θ be any angle in standard position and let P = (x,y) be a point on the terminal side of θ. If r = is the from (0,0) to (x,y), the six trigonometric functions of θ are defined by the following ratios: Since P = (x,y) is any point on the terminal side of θ other than the origin, r = cannot be zero. (Think of r as the radius.distance from the origin to P) Ex 1: Let P = (1,-3) be a point on the terminal side of θ. Find each of the six trigonometric functions of θ. (Start by drawing the angle in standard position.)

12 Trigonometric Functions of Quadrantal Angles: Ex. 2: Find the values of the six trigonometric functions, if they exist, for each of the four quadrantal angles. a) θ = 0 = 0 b) θ = 90 = π 2 c) θ = 180 = π d) θ = 270 = 3π 2 The Signs of the Trigonometric Functions: If θ is not a quadrantal angle, the sign of a trigonometric function depends on the quadrant in which θ lies. In all four quadrants, r, is positive (radius). Since sinθ = y, the sign of this ratio depends on y. Similarly the sign of the cosine ratio depends on x. Tangent is r the ratio of sine and cosine so it is dependent on the signs of x and y. *Think alphabetical (x,y) (cosine,sine) Now, look at all 4 quadrants: Your book s phrase: A Smart Trig Class (see pg. 517) Ex 3: If sinθ < 0 and cosθ < 0, name the quadrant in which angle θ lies.

13 Ex. 4: Given tanθ = 1 and cosθ < 0, find sinθ and secθ. Reference Angles: Let θ be a nonacute angle in standard position that lies in a quadrant. Its is the angle θ ( ) formed by the terminal side of θ and the x-axis. a) θ is in Quadrant II b) θ is in Quadrant III c) θ is in Quadrant IV Ex. 5: Find the reference angle, θ, for each of the following angles: a) θ = 210 b) θ = 7π 4 c) θ = 240 d) θ = 3.6 Finding Reference Angles for Angles Greater than 360 (2π) or less than 360 ( 2π): 1) Find a positive angle α less than 360 or 2π that is with the given angle. 2) Draw α is standard position. 3) Use the drawing to find the reference angle for the given angle. The positive acute angle formed by the terminal side of α and the x-axis is the.

14 Ex. 6: Find the reference angle for each of the following angles: a) θ = 665 b) θ = 15π 4 c) θ = 11π 3 Evaluating Trigonometric Functions Using Reference Angles: The value of a trigonometric function of any angle θ is found as follows: 1. Find the associated reference angle, θ, and the function value for θ. 2. Use the quadrant in which θ lies to prefix the appropriate sign to the function value in step 1. Ex. 7: Use reference angles to find the exact value of the following trigonometric functions: 1. sin300 b) tan 5π 4 c) sec ( π 6 ) Ex. 8: Use reference angles to find the exact value of each of the following trigonometric functions: a) cos 17π b) sin ( 22π ) 6 3 Homework: pg. 525 #3-84 (multiples of 3)---Use page 524 as a reference.

15 5.4: Trigonometric Functions of Real Numbers; Periodic Functions Date: Pre-Calculus Unit Circle: The equation of this unit circle is:. Recall: s = rθ, so in the unit circle, where the central angle measures t radians, s =. So, in a unit circle, the radian measure of the central angle is equal to the length of the intercepted arc. x =, y = Definitions of the Trigonometric Functions in Terms of a Unit Circle If t is a real number and P=(x,y) is a point on the unit circle that corresponds to t, then sint = csct =, y 0 cost = sect =, x 0 tant =, x 0 cott =, y 0 Note: x 2 + y 2 = 1 is equivalent to the Pythagorean identity cos 2 x + sin 2 x = 1. Ex. 1: Find the values of the trig functions at t if P = ( 3 2, 1 2 ). Ex. 2: Use a unit circle to find the values of the trigonometric functions at t = π.

16 Domain and Range of Sine and Cosine Functions The domain of the sine function and the cosine function is (, ), the set of all real numbers. The range of these functions is [ 1,1], the set of all real numbers from -1 to 1, inclusive. Even and Odd Trigonometric Functions Even: f( t) = Odd: f( t) = If P(cost,sint); Q((cos(-t),sin(-t)) The x-coordinates of P and Q are the same. Thus, cos( t) =, so cosine is an function. The y-coordinates of P and Q are opposites of each other. Thus sin( t) =, so sine is an function. EVEN FUNCTIONS: ODD FUNCTIONS: Ex. 3: Find the exact value of each trigonometric function: a) cos ( 60 ) b) tan ( π 6 )

17 A function f is periodic if there exists a positive number p such that f(t + p) = f(t) for all t in the domain of f. The smallest positive number p for which f is periodic is called the period of f. Periodic Properties of the Sine and Cosine Functions: sin(t + 2π) = sint and cos(t + 2π) = cost The sine and cosine functions are periodic functions and have a period of. (Similarly, cosecant and secant have a period of 2π.) Ex. 4: Find the exact value of each trigonometric function: a) cos405 b) sin 7π 3 Periodic Properties of the Tangent and Cotangent Functions tan(t + π) = tant and cot(t + π) = cotπ The tangent and cotangent functions are periodic functions and have period Repetitive Behavior of the Sine, Cosine, and Tangent Functions: For any integer n and real number t, sin(t + 2πn) = cos(t + 2πn) = tan(t + πn) = Homework: pg. 532 #1 18, 20 36(e), 44 46

18

19 5.5: Graphs of Sine and Cosine Date: Pre-Calculus Intro to Sine and Cosine Functions: y = sinx Amplitude: Period: Scaling: y = cosx Amplitude: Period: Scaling: Graphing sine and cosine functions in general: y = Asin(Bx C) + D and y = Acos(Bx C) + D 1. A = amplitude (If A<0, the graph is reflected over the x-axis.) 2. Period = 2π B 3. Horizontal shift (called the ): The starting point changes from x = 0 to x = C B (Find this by setting Bx C = 0.) If number C B is called the phase shift. C > 0, the shift is to the right. If C < 0, the shift is to the left. The B B 4. Vertical shift: The constant D causes vertical shifts in the graph. If D is positive, the shift is D units. If D is negative, the shift is D units. These vertical shifts result in graphs that oscillate about the horizontal line rather than the x-axis. Thus, the maximum y is D + A and the minimum y is D - A. (These values give the of the function.)

20 Examples: Graph each function. Draw two cycles. Label the max/min points and intercepts (5 key points per cycle). Ex 1: y = 3cos (x + π) Amplitude: Period: Scale by: Vertical shift: Phase shift: Ex 2: y = 2 sin ( 2π x) 3 Amplitude: Period: Scale by: Vertical shift: Phase shift:

21 Ex 3: y = 2 cos 3 (x π ) 2 4 Amplitude: Period: Scale by: Vertical shift: Phase shift: Ex 4: y = 3 cos(x + π) 3 Amplitude: Period: Scale by: Vertical shift: Phase shift: Homework: pg. 554 #13, 20, 23, 26, 41, 44, 47, 50, 62, 64, 66, 84 87

22

23 5.6: Graphs of Other Trig Functions (Tangent and Cotangent) Date: Pre-Calculus y = tan x Period: VA: Domain: Range: x-intercepts: Symmetry: Graphing Variations of y = tan x in the form y = A tan(bx C) 1. Period = 2. Find two consecutive asymptotes by finding an interval containing one period: The asymptotes occur at: a. b. 3. Identify an x-intercept, midway between the consecutive asymptotes 4. Amplitude = Ex 1: y = tan (π x) Amplitude: Period: Asymptotes:

24 y = cotx Period: VA: Domain: Range: x-intercepts: Symmetry: Graphing Variations of y = cot x in the form y = A cot(bx C) 1. Period = 2. Find two consecutive asymptotes by finding an interval containing one period: The asymptotes occur at: a. b. 3. Identify an x-intercept, midway between the consecutive asymptotes 4. Amplitude = Ex 2: y = cot ( x ) 2 Amplitude: Period: Asymptotes:

25 Ex 3: y = 2cot (x + π ) 2 Amplitude: Period: Phase Shift: Asymptotes: Ex 4: y = 1 cot 2x 2 Amplitude: Period: Asymptotes: Ex 5: y = 2 tan 1 x 2 Amplitude: Period: Asymptotes: Homework: pg. 567 #1 4, 6 12(e), 13 16, 18 24(e), 45, 46, 59

26

27 5.6 Continued: Graphs of Other Trigonometric Functions Date: Pre-Calculus y = sec x y = cscx Reciprocal function: Domain: Range: Period: Vertical Asymptotes: Symmetry: y = sec x y = csc x

28 Ex 1: y = 3 csc x 2 Amplitude: Period: Scale by: Ex 2: y = 1 sec x 2 Amplitude: Period: Scale by: Ex 3: y = csc x 3 Amplitude: Period: Scale by:

29 Ex 4: y = sec(πx) 1 Amplitude: Period: Scale by: Ex 5: y = 2 csc (x + π 4 ) Amplitude: Period: Scale by: Phase shift: Homework: pg. 568 #26 44(e), 47, 48, 61, 93 96

30

31 5.7: Inverse Trigonometric Functions Date: Pre-Calculus Recall: In order for a function to have an inverse function, the graph of the function must pass the. On their entire domain, the trig functions do pass this test. INVERSE SINE Restrict y = sinx so that. Then it will have an inverse function. The inverse function for sinx is or. (For inverse functions, the domain/range are switched.) Sine is an function - symmetry with the. Note: y = sin 1 x 1 sinx. Graphically inverses are reflections over the line. (true for exp/log functions) θ sinθ Ex 1: Find the exact value of each of the following: a) sin 1 ( 3 2 ) = b) sin 1 ( 2 2 ) =

32 INVERSE COSINE Restrict y = cosx so that. Then it will have an inverse function. The inverse function for cosx is or. (For inverse functions, the domain/range are switched.) Cosine is an function - symmetry with the. θ cosθ Ex 2: Find the exact value of each of the following: a) cos 1 ( 1 2 ) = b) cos 1 ( 2 2 ) = INVERSE TANGENT Restrict y = tanx so that. Then it will have an inverse function. The inverse function for tanx is or. (For inverse functions, the domain/range are switched.) Cosine is an function - symmetry with the. θ tanθ

33 Ex. 3: Find the exact value of each of the following: a) tan 1 ( 1) = b) tan 1 ( 3 3 ) = Ex. 4: Use a calculator to find the value of each expression to four decimal places. a) cos 1 ( 1 3 ) = b) tan 1 (35.85) = c) sin 1 2 = COMPOSITION OF FUNCTIONS INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS Recall: f(f 1 (x)) = x for and f 1 (f(x)) = x for. INVERSE PROPERTIES sin(sin 1 x) = x for sin 1 (sinx) = x for cos(cos 1 x) = x cos 1 (cosx) = x tan(tan 1 x) = x tan 1 (tanx) = x for for for for Ex. 5: Find the exact value (if possible) for each of the following expressions: a) cos(cos 1 0.7) = b) sin 1 (sinπ) = c) cos[cos 1 ( 1.2)] = EVALUATING A COMPOSITE TRIG EXPRESSION (Draw a triangle!) Ex. 6: Find the exact value of sin (tan 1 ( 3 4 )).

34 Ex. 7: Find the exact value of cos [sin 1 ( 1 2 )]. SIMPLIFYING EXPRESSIONS INVOLVING TRIG FUNCTIONS Ex. 8: If x > 0, write sec(tan 1 x) as an algebraic expression in x. Homework due Tuesday 12/13: Graphing Trig Functions Review (handout) & pg. 584 #1 30 (no calc for #1 18) Quiz #8 ( ) NON CALCULATOR! Wednesday 12/14 Homework due Wednesday 12/14 (Section 5.7): pg. 584 #32 72(e), 95, 96, , Make sure you know the sin, cos, tan values in the tables!

35 5.8: Applications of Trigonometric Functions Date: Pre-Calculus Objectives: Solve a right triangle. Solve problems involving bearings. Solving Right Triangles finding the missing lengths of the sides and angles of a triangle. Generic diagram of a right triangle Ex. 1: Let A = 62.7 and a = 8.4. Solve the right triangle, rounding lengths to two decimal places. Ex. 2: from a point on level ground 80 feet from the base of the Eiffel Tower, the angle of elevation is Approximate the height of the Eiffel Tower to the nearest foot.

36 Ex. 3: A guy wire (support wire) is 13.8 yards long and is attached from the ground to a pole 6.7 yards above the ground. Find the angle, to the nearest tenth of a degree that the wire makes with the ground. Using Two Right Triangles to Solve a Problem Ex. 4: You are standing on level ground 800 feet from Mt. Rushmore, looking at the sculpture of Abraham Lincoln s face. The angle of elevation to the bottom of the sculpture is 32 and the angle of elevation to the top is 35. Find the height of the sculpture of Lincoln s face to the nearest tenth of a foot. Trigonometry and Bearings The bearing from a point O to a point P is the acute angle, measured in degrees between OP and a north-south line. Ex. 5: Find the bearing from O to P in each diagram.

37 Ex. 6: A boat leaves the entrance to a harbor and travels 150 miles on a bearing of N 53 E. How many miles north and how many miles east form the harbor has the boat traveled? Ex. 7: You leave your house and run 2 miles due west followed by 1.5 miles due north. At that time, what is your bearing from your house? Ex. 8: You leave the entrance to a system of hiking trails and hike 2.3 miles on a bearing of S 31 W. Then the trail turns 90 and you hike 3.5 miles on a bearing of N 59 W. At that time: a) How far are you, to the nearest tenth of a mile, from the entrance to the trail system? b) What is your bearing, to the nearest tenth of a degree, from the entrance to the trail system? Homework: pg. 595 #6, 10, 14, 16, 42 58(e) (We will continue section 5.8 tomorrow.)

38

39 5.8 Continued: Simple Harmonic Motion Date: Pre-Calculus Simple Harmonic Motion Consider a ball attached to a spring hung from the ceiling. You pull the ball down and then release it. If we neglect the effects of friction and air resistance, the ball will continue bobbing up and down on the end of the spring. These up-and down oscillations are called simple harmonic motion. The position of the ball before you pull the ball down is d=0. This rest position is called the equilibrium position. An object that moves on a coordinate axis is in simple harmonic motion if its distance from the origin, d, at time t, is given by either d = acos(ωt) or d = asin (ωt) Use d = acos(ωt) if Use d = asin (ωt) if The motion has a, the maximum displacement of the object from its rest position. The of the motion is 2π, where ω > 0. ω The period gives the time it takes for the motion to go through one. Ex. 10: A ball on a spring is pulled 6 inches below its rest position and then released. The period for the motion is 4 seconds. Write the equation for the ball s simple harmonic motion.

40 Frequency of an Object in simple Harmonic Motion An object in simple harmonic motion given by d = acos(ωt) or d = asin (ωt) has frequency f, given by f = ω, ω > 0. 2π Equivalently, f = 1 period. Ex. 11: An object moves in simple harmonic motion described by d = 12cos π t, where t is measured in seconds 4 and d in centimeters. Find: a) the maximum displacement: b) the frequency: c) the time required for one cycle: Ex. 12: An object moves in simple harmonic motion described by d = 1 sin 2 (π t π ), where t is measured in 4 2 seconds and d in centimeters. Find: a) the maximum displacement: b) the frequency: c) the time required for one cycle: d) the phase shift of the motion: Homework: pg. 596 #20, 24, 26, 37, 39, 60, 62, 70, 72 77; pg. 601 #18 20, pg. 603 # (e)