I IV II III 4.1 RADIAN AND DEGREE MEASURES (DAY ONE) COMPLEMENTARY angles add to90 SUPPLEMENTARY angles add to 180

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1 4.1 RADIAN AND DEGREE MEASURES (DAY ONE) TRIGONOMETRY: the study of the relationship between the angles and sides of a triangle from the Greek word for triangle ( trigonon) (trigonon ) and measure ( metria) (metria ANGLE OF ROTATION: an angle formed by a ray that is rotated around its endpoint INITIAL SIDE: starting position TERMINAL SIDE: final position STANDARD POSITION: when the initial side lies along the positive x-axis and its endpoint is at the origin COTERMINAL ANGLES: angles with the same terminal side 4 terminal side vertex initial side STANDARD POSITION 5 DEGREE MEASURE 90 < θ < 180 II III 180 < θ < 70 0 < θ < 90 I IV 70 < θ < EXAMPLE #1 Find two co-terminal angles (in degrees), one positive and one negative for each. A. θ = 180 B. θ = 7 C. θ = 15 COMPLEMENTARY angles add to90 SUPPLEMENTARY angles add to α+β=90 4 α+β=180 β α β α 5 - Complementary angles - Supplementary angles

2 EXAMPLE # Find the complement and supplement of each angle, if possible. A. θ = 160 B. θ = 54 C. θ = 97 DEFINITION OF A RADIAN One RADIAN is the measure of a central angle( θ ) that intercepts an arc( s) equal in length to the radius( r) of the circle. Circumference of a circle: C =r Full rotation: radians 1 radian r θ s = r 3 radians r 6 radians 4 radians 5 radians ANY GIVEN ANGLE θ HAS INFINITELY MANY COTERMINAL ANGLES ±, θ n where n is any integer CONVERSION OF ANGLE MEASURES radians 1. To convert DEGREES to RADIANS # degrees To convert RADIANS to DEGREES # radians radians 0 = 90 = 180 = 70 = 360 =

3 EXAMPLE #3 Convert from radians degrees or degrees radians. A. 135 B. 70 C. rad D. rad 3 EXAMPLE #4 Find two co-terminal angles (in radians), one positive and one negative for each. A. 7 θ = B θ = C. 6 θ = 3 EXAMPLE #5 Find the complement and supplement of each angle. A. θ = 5 7 B. θ = 3

4 4.1 RADIANS AND DEGREE MEASURES (DAY TWO) APPLICATIONS DEFINITION OF ARC LENGTH To convert angle measures, use the following conversion ARC LENGTH or CIRCULAR LENGTH is s = rθ, where θ is in radians. The formula for arc length can be used to analyze the motion of a particle moving at a CONSTANT SPEED along a circular path. LINEAR AND ANGULAR SPEED Consider a particle moving at a constant speed along a circular arc of radiusr. Ifs is the length of the arc traveled in timet, them the LINEAR SPEED of the particle is arc length s LINEAR SPEED = = time t In addition, ifθ is the angle (in radian measure) corresponding to the arc lengths, the ANGULAR SPEED of the particle is central angle θ ANGULAR SPEED = = time t EXAMPLE #1 The second hand of a clock is 10. centimeters long. Find the linear speed of the tip of the second hand. EXAMPLE # A 15-inch diameter tire on a car makes 9.3 revolutions per second. A. Find the angular speed of the tire in radian per second. B. Find the linear speed of the car.

5 LATITUDE and LONGITUDE geographic coordinates are often expressed as degrees, minutes, and seconds. Any decimal degree measure can be converted to this form and back. CONVERSION OF DEGREE DECIMAL D M'S" 1. D M'S" to DEGREE DECIMAL : 1 = 60 M' = 3600 S" 1 ' 3" = 1 + ' + 3" = = DEGREE DECIMAL to D M'S" : 1.34 = = 14.04= 14' =.4= " 1 14' " EXAMPLE #3 Convert the angle measure to decimal degree form. A. 19 '13" B " EXAMPLE #4 Convert the angle measure to D M' S " form. A B

6 4. TRIGONOMETRIC FUNCTIONS: THE UNIT CIRCLE THE UNIT CIRCLE As the real number line is wrapped around the UNIT CIRCLE, each real numbert corresponds to a point( x, y) on the circle. The ( x, y ) coordinate points in the first quadrant are directly related to the special right triangle theorems. REMEMBER SPECIAL RIGHT TRAINGLES sin 45 = cos 45 = tan 45 = sin 30 = sin 60 = cos 30 = cos 60 = tan 30 = tan 60 = TRIGONOMETRIC FUNCTIONS 1 csct 1 sect TRIGONOMETRIC FUNCTIONS sin t = y 1 csc t =, y y 0 cos t = x 1 sec t =, x x 0 1 sint 1 cost 1 cott tan t = y, x 0 x cot t = x, y 0 y 1 tant EXAMPLE #1 True or false. A. = 1 csct x B. cot 0 is undefined C. cot = 1

7 EXAMPLE # Evaluate the six trigonometric functions. A. t = 5 sint = csct = cost = sect = tant = cott = B. t = 7 6 sint = csct = cost = sect = tant = cott = DOMAIN AND PERIOD OF SINE AND COSINE f( x) = sin( x) - -1 f( x) = cos( x) SINE COSINE (, + ) Domain (, + ) Range 1, 1 DEFINITION OF A PERIODIC FUNCTION A function is PERIODIC if there exists a positive real numberc such that f t + c = f t ( ) ( ) for allt in the domain off. The smallest numberc for whichf is periodic is called the PERIOD. THE PERIOD OF SINE AND COSINE IS SINCE sin t + n = sin t and cos t + n = cos t ( ) ( ) ( ) ( )

8 EXAMPLE #3 Evaluate the trigonometric function. 3 A. 13 sin 6 B. 8 cos 3 EVEN AND ODD TRIGONOMETRIC FUNCTIONS 1. COSINE and SECANT are EVEN functions cos t = cos t and sec t = sec t ( ) ( ) ( ) ( ). SINE, COSECANT, TANGENT and COTANGENT are ODD functions sin t = sin t and csc t = csc t ( ) ( ) ( ) ( ) tan( t) = tan( t) and cot( t) = cot( t) EVALUATING TRIGONOMETRIC FUNCTIONS WITH A CALCULATOR You may want to rewrite expressions involving csc θ, sec θ, and cotθ in terms of sin θ, cos θ, and tanθ EXAMPLE #4 Evaluate the trigonometric function with a calculator. A. sin 3 B. cot( 1.5 ) C. cos( 1. ) D. 3 tan 4 E. csc(.1 ) F. sec(.8 )

9 4.3 RIGHT TRIANGLE TRIGONOMETRY (DAY ONE) THE SIX TRIGONOMETRIC FUNCTIONS sine cosecant cosine secant tangent cotangent RIGHT TRIANGLE DEFINITIONS OF TRIGONOMETRIC FUNCTIONS Letθ be an acute angle of a right triangle. Then the six trigonometric functions of angleθ are defined as follows: opp hyp sinθ = cscθ = hyp opp adj hyp cosθ = secθ = hyp adj opp adj tan θ = cotθ = adj opp REMEMBER: Soh-Cah-Toa EXAMPLE #1 True or false. 1 A. secθ = cscθ B. sin 45 = 1 C. = 10 3 rad

10 EXAMPLE # A right triangle has an adjacent leg of length and an opposite leg of length 5. Find the exact value of each of the six trigonometric functions. sin θ = csc θ = cos θ = sec θ = tan θ = cot θ = COFUNCTION IDENTITIES Ifθ is an acute angle, then the following are true: sin( 90 θ) = cosθ cos θ = sinθ tan( 90 θ) = cotθ cot θ = tanθ sec( 90 θ) = cscθ csc θ = secθ 1 EXAMPLE: sin 30 = = cos 60 This occurs because30 and 60 are COMPLEMENTARY ANGLES. COFUNCTIONS OF COMPLEMENTARY ANGLES ARE EQUAL EXAMPLE #3 Find each value of θ in degrees ( 0 < θ< 90 ) and radians 0 < θ< without using a calculator. A. 3 sinθ= B. tanθ= 3 C. secθ=

11 4.3 RIGHT TRIANGLE TRIGONOMETRY (DAY TWO) TRIGONOMETRIC IDENTITIES RECIPROCAL TRIGONOMETRIC IDENTITIES sinθ = cosθ = tanθ = cscθ secθ cotθ cscθ = secθ = cotθ = sinθ cosθ tanθ QUOTIENT TRIGONOMETRIC IDENTITIES sinθ cosθ tanθ = cotθ = cosθ sinθ PYTHAGOREAN TRIGONOMETRIC IDENTITIES sin θ + cos θ = 1 tan θ + 1= sec θ cot θ + 1= csc θ θ = θ θ = θ θ = θ sin 1 cos tan sec 1 cot csc 1 cos θ = 1 sin θ 1= sec θ tan θ 1= csc θ cot θ θ θ θ θ tan sec = 1 cot csc = 1 EXAMPLE #1 Letθ be an acute angle such that sinθ = 0.6. Find tanθ. EXAMPLE # Use trigonometric identities to transform one side of the equation into the other. cos sec 1 B. ( θ θ)( θ θ) A. θ θ = sec + tan sec tan = 1

12 C. cscθ sinθ = cot θ sinθ APPLICATIONS INVOLVING RIGHT TRIANGLES ANGLE OF ELEVATION is the angle from the horizontal UPWARD. ANGLE OF DEPRESSION is the angle from the horizontal DOWNWARD. EXAMPLE #3 A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as71.5. How tall is the tree? 50 ft 71.5 EXAMPLE #4 A person is 00 yards from a river. Rather than walk directly to the river, the person walks 400 yards along a straight path to the river s edge. Find the acute angleθ between this path and the river s edge. 00 yds 400 yds EXAMPLE #5 A 10 foot ladder leans against the side of a house. The ladder makes an angle of 60 with the ground. How far up the side of the house does the ladder reach?

13 4.4 TRIGONOMETRIC FUNCTIONS OF ANY ANGLE Using the lengths of the sides of the triangle, we can form six ratios that define the six trigonometric functions of the acute angleθ. DEFINITION OF TRIGONOMETRIC FUNCTIONS OF ANY ANGLE Letθ be an angle in standard position with( x, y) a point on the terminal side ofθ and r = x + y 0. y r sinθ = csc θ =, y 0 r y x r cosθ = sec θ =, x 0 r x y x tan θ =, x 0 cot θ =, y 0 x y (x,y) r θ EXAMPLE #1 Let( 3, 4) be a point on the terminal side ofθ. Find the sine, cosine, and tangent ofθ. The SIGNS of the trigonometric functions in the four quadrants can be determined from the definitions of y the functions sinθ = y cosθ = x tan θ =, etc. x EXAMPLE # Given sinθ = 3 5 and cos θ < 0. Find tan θ and sec θ.

14 REFERENCE ANGLES The values of the trigonometric functions of angles greater than 90 (or less than 0 ) can be determined from their values at corresponding acute angles called REFERENCE ANGLES. DEFINITION OF REFERENCE ANGLES Letθ be an angle in standard position. Its REFERENCE ANGLE is the acute angleθ 'formed by the TERMINAL SIDE ofθ and the HORIZONTAL AXIS. Quadrant II reference angle θ' θ reference angle θ' θ Quadrant III EXAMPLE #3 Find the reference angle. A. θ = 300 B. θ = 4.8 C. θ = 145 D. θ = 14 3 TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS EVALUATING TRIGONOMETRIC FUNCTIONS OF ANY ANGLE To find the value of a trigonometric function of any angleθ : 1. Determine the function value for the associated reference angleθ '.. Depending on the quadrant in whichθ lies, assign the appropriate sign to the function value. EXAMPLE #4 Evaluate the trigonometric function without a calculator. A. 4 cos B. ( ) 3 tan 10 C. 11 csc 4

15 EXAMPLE #5 Find sinθ given that cotθ = 3andθ is in Quadrant II. 3 EXAMPLE #6 Use a calculator to approximate two values ofθ that satisfy the equation. A. cotθ = θ 360 B. cscθ = θ C. secθ = θ

16 4.5 GRAPHS OF SINE/COSINE FUNCTIONS (DAY ONE) BASIC SINE AND COSINE CURVES y = sin x amplitude period = = y = cosx amplitude period = = SINE is symmetric with respect to the ORIGIN. COSINE is symmetric with respect to the Y-AXIS. There are 5 key points in one period of each graph that include: intercepts maximums minimums Find these key points on the sketches above. AMPLITUDE AND PERIOD ALL OF THE RULES THAT WE LEARNED FOR REFLECTING, STRETCHING/COMPRESSING AND SHIFTING IS STILL TRUE FOR SINE AND COSINE CURVES DEFINITION OF AMPLITUDE OF SINE AND COSINE CURVES The AMLITUDE of y = a sin x and y = a cos x represents HALF the distance between the MAXIMUM and MINIMUM values of the function and is given by Amplitude = a 1 EXAMPLE #1 Sketch y = sin x and y = 3sin x in the same coordinate plane. Be sure to include full periods

17 PERIOD OF SINE AND COSINE FUNCTIONS PERIOD OF SINE AND COSINE FUNCTIONS Letb be a positive real number. The PERIOD of y = a sin bx and y = a cos bx is given by Period = b HORIZONTAL STRETCH happens if 0< b < 1 since the period is GREATER than. HORIZONTAL COMPRESSION happens if b > 1 since the period is LESS than. x y and y x in the same coordinate plane. EXAMPLE # Sketch = cos = cos( ) EXAMPLE #3 Sketch y = ( ) 3 sin 4 x in the coordinate plane

18 4.5 GRAPHS OF SINE/COSINE FUNCTIONS (DAY TWO) TRANSLATIONS OF SINE AND COSINE CURVES THE PHASE SHIFT IS THE AMOUNT THE GRAPH IS SHIFTED RELATIVE TO THE PERIOD GRAPHS OF SINE AND COSINE FUNCTIONS The graphs of = sin( ) = cos( ) y a bx c and y a bx c have the following characteristics. (Assume b > 0 ) Amplitude = a Period = b The LEFT and RIGHT ENDPOINTS of a one-cycle interval can be determined be solving the equations bx c = 0 and bx c = c x c + b b b where b is the new period and c is the PHASE SHIFT. b 1 1 EXAMPLE #1 Sketch y = sin x and y = sin x in the same coordinate plane EXAMPLE # Sketch the graph of = 3cos( + 4) y x

19 The final type of transformation is the VERTICAL SHIFT that happens when we add some numberd to the equation y = d + a sin bx c and y = d + a cos bx c ( ) ( ) EXAMPLE #3 Sketch the graph of = + 3cos( ) y x

20 4.6 GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS GRAPH OF THE TANGENT FUNCTION y = tan x Amplitude = Period = Since the PERIOD of y = tan x is, all vertical asymptotes occur when x = + n, wheren is any integer. Sketching the graph of y = a tan( bx c) is similar to sketching that graph y a sin( bx c) Starting and ending points: x EXAMPLE #1 Sketch the graph of y = tan. bx c = and bx c = = Period = b EXAMPLE # Sketch y 3 tan( x) =

21 GRAPH OF THE COTANGENT FUNCTION y = cotx Amplitude = Period = Since the PERIOD of y = cot x is, all other vertical asymptotes occur when x = 0+ n, wheren is any integer. x EXAMPLE #3 Sketch y = cot

22 GRAPHS OF THE RECIPROCAL FUNCTIONS 3 The easiest way to sketch the graph of secant or cosecant is to sketch the graph of cosine or sine first. 1 1 csc x = and sec x = sin x cos x Like the tangent function, sec x has vertical asymptotes when x = + n. And like the cotangent function, csc x has vertical asymptotes when x = 0+ n. y = cscx Period = y = secx Period = EXAMPLE #4 Sketchy = sin x and y csc x + = + in the same coordinate plane

23 4 EXAMPLE #5 Sketch y cos( x) and y sec( x) = = in the same coordinate plane DAMPED TRIGONOMETRIC FUNCTIONS The PRODUCT of two functions can be graphed using the properties of the individual functions. For example, f ( x) = x sin x is the product of y = x and y = sin x. The graph of f ( x) = x sin x lies between the lines y = x and y = x. The graph of sin x takes on the MAXIMUM and MINIMUM values of the lines y = x and y = x As you can see from the graph above f ( x) = x sin x =± x at x = + n f ( x) = x sin x = 0 at x = n maximum / minimum intercept In the function f ( x) = x sin x, the factorx is called the DAMPING FACTOR.

24 4.7 INVERSE TRIGONOMETRIC FUNCTIONS INVERSE SINE FUNCTION (ARCSINE) Remember: only ONE-TO-ONE functions have inverses. 1 y = sin x If we RESTRICT the domain to be the closed interval x then 1. On the interval,, the function = sin y x is INCREASING.. On the interval,, y = sin x takes on its FULL RANGE of values. 3. On the interval,, y = sin x passes the HORIZONTAL LINE TEST. DEFINITION OF INVERSE SINE FUNCTION The INVERSE SINE FUNCTION is defined by y = arcsin x if and only if sin y = x where 1 x 1 and x. The DOMAIN of y = arcsin x is 1, 1 and the RANGE is,. 1 y = arcsin x OR y = sin x The ARCSINE ofx is the ANGLE (or NUMBER) whose SINE isx EXAMPLE #1 Find the exact value. A. 1 arcsin B. 1 3 sin 1 C. sin ( ) D. arcsin

25 OTHER INVERSE TRIGONOMETRIC FUNCTIONS The cosine function is DECREASING on the interval 0 y and passes the HORIZONTAL LINE TEST. 1 y = cos x y = arccos x OR y = cos x You can define INVERSE TANGENT in a similar way by restricting the domain of y = tan x to the interval < y <. = arcsin = sin 1 y x x = arccos = cos 1 y x x = arctan = tan 1 y x x Domain :[ 1, 1 ] Range :, :[ 1, 1 ] :[ 0, ] Domain Range Domain : (, + ) Range :, DEFINITION OF INVERSE TRIGONOMETRIC FUNCTIONS Function Domain Range y = arcsin x if and only if sin y = x 1 x 1 y y = arccos x if and only if cos y = x 1 x 1 0 y y = arctan x if and only if tan y = x < x <+ < y <

26 EXAMPLE # Find the exact value. 3 A. arccos 1 1 B. cos ( 3 ) C. tan ( 1 ) E. ( ) arctan 0 F. arccos 3 EXAMPLE #3 Approximate the value (if possible). 1 1 A. arctan 3. B. arcsin C. cos ( ) D. tan ( 8.45) COMPOSITIONS OF FUNCTIONS Earlier in the semester we learned that INVERSE FUNCTIONS posses the properties ( 1 f f ( x) ) = x and f 1 ( f ( x) ) = x INVERSE PROPERTIES If 1 x 1 and y, then 1 sin arcsin x = x and sin sin y = y ( ) ( ) If 1 x 1 and 0 y, then 1 ( ) arccos( cos ) cos cos x = x and y = y If x is a real number and < y <, then 1 tan arctan x = x and tan tan y = y ( ) ( ) These inverse properties do NOT apply for ALL values of x and y. For example 5 5 arcsin sin = arcsin = arccos cos arccos = = 4 4 4

27 4 EXAMPLE #4 Find the exact value (if possible). A. tan( arctan( 5 )) C. 5 arcsin sin 3 B. cos cos 1 7 D. 11 arctan tan 6 EXAMPLE #5 Find the exact value of the expression. A. tan arccos B csc tan 5 x EXAMPLE #6 Find an expression for tan arcsin. 5

28 4.8 APPLICATIONS AND MODELS APPLICATIONS EXAMPLE #1 At a point 00 feet from the base of a building, the angle of elevation to the bottom of a smokestack is35, and the angle of elevation to the top is53. Find the heights of the smokestack alone. s a EXAMPLE # A large helium-filled penguin is moored at the beginning of a parade route awaiting the start of the parade. Two cables attached to the underside of the penguin make angles of 48 and 40 with the ground and are in the same plane as perpendicular line from the penguin to the ground. If the cables are attached to the ground 10 feet from the other, how high above the ground is the penguin? EXAMPLE #3 From the top of a 100-foot tall building a man observes a car moving toward the building. If the angle of depression of the car changes from to 46 during the period of observation, how far does the car travel?

29 HARMONIC MOTION SIMPLE HARMONIC MOTION can be described by a SINE or COSINE curve. DEFINITION OF SIMPLE HARMONIC MOTION A point that moves on a coordinate line is said to be in SIMPLE HARMONIC MOTION if its distanced from the origin at timet is given by either d = a sinωt or d = a cosωt wherea and ω are real numbers such thatω > 0. The motion has amplitude a, period ω, and frequency ω. PERIOD IS ω FREQUENCY ω t EXAMPLE #7 Given the equation for simple harmonic motion d = 3sin, find: A. the maximum displacement, B. the frequency of the simple harmonic motion, and C. the period of the simple harmonic motion.