Section 5.1 Angles and Radian Measure. Ever Feel Like You re Just Going in Circles?

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1 Section 5.1 Angles and Radian Measure Ever Feel Like You re Just Going in Circles? You re riding on a Ferris wheel and wonder how fast you are traveling. Before you got on the ride, the operator told you that the wheel completes two full revolutions every minute and that your seat is 5 feet from the center of the wheel. You just rode on the merry-go-round, which made.5 complete revolutions per minute. Your wooden horse was 0 feet from the center, but your friend, riding beside you was only 15 feet from the center. Were you and your friend traveling at the same rate? In this section, we study both angular speed and linear speed and solve problems similar to those just stated. Objective #1: Recognize and use the vocabulary of angles. Solved Problem #1 1a. True or false: When an angle is in standard position, its initial side is along the positive y-axis. False; When an angle is in standard position, its initial side is along the positive x-axis. Pencil Problem #1 1a. True or false: When an angle is in standard position, its vertex lies in quadrant I. 1b. Fill in the blank to make a true statement: If the terminal side of an angle in standard position lies on the x-axis or the y-axis, the angle is called a/an angle. 1b. Fill in the blank to make a true statement: A negative angle is generated by a rotation. Such an angle is called a quadrantal angle. Solved Problem #. Fill in the blank to make a true statement: An angle that is formed by 1 of a complete rotation measures degrees and is called a/an angle. Such an angle measures 180 degrees and is called a straight angle. Objective #: Use degree measure. Pencil Problem #. Fill in the blank to make a true statement: An angle that is formed by 1 of a complete rotation 4 measures degrees and is called a/an angle. Copyright 014 Pearson Education Inc. 16

2 Algebra and Trigonometry 5e Solved Problem #. A central angle, θ, in a circle of radius 1 feet intercepts an arc of length 4 feet. What is the radian measure of θ? The radian measure of the central angle, θ, is the length of the intercepted arc, s, divided by the radius s of the circle, r: θ. In this case, s = 4 feet and r r = 1 feet. s 4 feet θ.5 r 1 feet The radian measure of θ is.5. Objective #: Use radian measure. Pencil Problem #. A central angle, θ, in a circle of radius 10 inches intercepts an arc of length 40 inches. What is the radian measure of θ? Objective #4: Convert between degrees and radians. Solved Problem #4 4a. Convert 60 to radians. To convert from degrees to radians, multiply by radians. Then simplify. 180 radians 60 radians 60 radians Pencil Problem #4 4a. Convert 15 to radians. Express your answer as a multiple of. 4b. Convert 00 to radians. radians 00 radians 5 00 radians b. Convert 5 to radians. Express your answer as a multiple of. 164 Copyright 014 Pearson Education Inc.

3 Section 5.1 4c. Convert radians to degrees. 4 To convert from radians to degrees, multiply by 180. Then simplify. radians radians 45 4 radians 4 4c. Convert radians to degrees. 4d. Convert 6 radians to degrees radians 4.8 radians 4d. Convert radians to degrees. Round to two decimal places. Objective #5: Draw angles in standard position. Solved Problem #5 5a. Draw the angle θ in standard position. 4 Since the angle is negative, it is obtained by a clockwise rotation. Express the angle as a fractional part of The angle θ is clockwise direction. of a full rotation in the Pencil Problem #5 5 5a. Draw the angle θ in standard position. 4 Copyright 014 Pearson Education Inc. 165

4 Algebra and Trigonometry 5e 5b. Draw the angle α in standard position. 4 Since the angle is positive, it is obtained by a counterclockwise rotation. Express the angle as a fractional part of. 4 8 The angle α 4 is 8 of a full rotation in the counterclockwise direction. 7 5b. Draw the angle α in standard position c. Draw the angle γ in standard position. 4 Since the angle is positive, it is obtained by a counterclockwise rotation. Express the angle as a fractional part of The angle γ is or 5 1 full rotation in the 8 counterclockwise direction. Complete one full 16 5c. Draw the angle γ in standard position. rotation and then 5 8 of a full rotation. 166 Copyright 014 Pearson Education Inc.

5 Solved Problem #6 6a. Find a positive angle less than 60 that is coterminal with a 400 angle. Since 400 is greater than 60, we subtract = 40 A 40 angle is positive, less than 60, and coterminal with a 400 angle. Objective #6: Find coterminal angles. Pencil Problem #6 6a. Find a positive angle less than 60 that is coterminal with a 95 angle. Section 5.1 6b. Find a positive angle less than that is coterminal with a angle. 15 6b. Find a positive angle less than that is coterminal with a angle. 50 Since is negative, we add A 9 angle is positive, less than, and 15 coterminal with a angle. 15 Copyright 014 Pearson Education Inc. 167

6 Algebra and Trigonometry 5e 6c. Find a positive angle less than that is coterminal with a 17 angle. 6c. Find a positive angle less than that is coterminal 1 with a angle. 7 Since 17 is greater than 4, we subtract two multiples of A 5 angle is positive, less than, and coterminal with a 17 angle. Solved Problem #7 7. 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. Then round your answer to two decimal places. We begin by converting 45 to radians. radians 45 radians 45 radians Now we use the arc length formula s rθ with the radius r = 6 inches and the angle θ radians. 4 6 s rθ (6 in.) in. in in. 4 4 Objective #7: Find the length of a circular arc. Pencil Problem #7 7. A circle has a radius of 8 feet. Find the length of the arc intercepted by a central angle of 5. Express arc length in terms of. Then round your answer to two decimal places. 168 Copyright 014 Pearson Education Inc.

7 Section 5.1 Objective #8: Use linear and angular speed to describe motion on a circular path. Solved Problem #8 8. 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. We are given the angular speed in revolutions per minute: ω = 45 revolutions per minute. We must express ω in radians per minute. 45 revolutions radians ω 1 minute 1 revolution 90 radians 90 or 1 minute 1 minute Now we use the formula υ rω in. υ rω 1.5 in. 44 in./min 1 min min Pencil Problem #8 8. A Ferris wheel has a radius of 5 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. Copyright 014 Pearson Education Inc. 169

8 Algebra and Trigonometry 5e Answers for Pencil Problems (Textbook Exercise references in parentheses): 1a. False 1b. clockwise. 90; right. 4 radians (5.1 #7) 5 4a. radians (5.1 #15) 4b. radians (5.1 #19) 4c. 90 (5.1 #1) 4 4 4d (5.1 #5) 5a. (5.1 #47) 5b. (5.1 #41) 5c. (5.1 #49) 6a. 5 (5.1 #57) 6b (5.1 #67) 6c (5.1 #69) ft 1.4 ft (5.1 #7) ft/min 14 ft/min (5.1 #98) 170 Copyright 014 Pearson Education Inc.

9 Section 5. Right Triangle Trigonometry Measuring Up, Way Up Did you ever wonder how you could measure the height of a building or a tree? How can you find the distance across a lake or some other body of water? In this section, we show how to model such situations using a right triangle and then using relationships among the lengths its sides and the measures of its angles to find distances that are otherwise difficult to measure. These relationships are known as the trigonometric functions. Objective #1: Use right triangles to evaluate trigonometric functions. Solved Problem #1 1a. Find the value of each of the six trigonometric functions of θ in the figure. Pencil Problem #1 1a. Find the value of each of the six trigonometric functions of θ in the figure. We first need to find c, the length of the hypotenuse. We use the Pythagorean Theorem. c a b c 5 5 We apply the definitions of the six trigonometric functions. Note that the side labeled a = is opposite angle θ and the side labeled b = 4 is adjacent to angle θ. opposite sinθ hypotenuse 5 adjacent 4 cosθ hypotenuse 5 opposite tanθ adjacent 4 hypotenuse 5 cscθ opposite hypotenuse 5 secθ adjacent 4 adjacent 4 cotθ opposite Copyright 014 Pearson Education Inc. 171

10 Algebra and Trigonometry 5e 1b. Find the value of each of the six trigonometric functions of θ in the figure. Express each value in simplified form. 1b. Find the value of each of the six trigonometric functions of θ in the figure. Express each value in simplified form. We first need to find b. a b c 1 b 5 1b 5 b 4 b 4 6 We apply the definitions of the six trigonometric functions. opposite 1 sinθ hypotenuse 5 adjacent 6 cosθ hypotenuse 5 opposite tanθ adjacent hypotenuse 5 cscθ 5 opposite 1 hypotenuse secθ adjacent adjacent 6 cotθ 6 opposite 1 17 Copyright 014 Pearson Education Inc.

11 Objective #: Find function values for 0, 6 45, 4 and 60. Solved Problem # a. Use the right triangle to find csc 45. Pencil Problem # Section 5. a. Use the right triangle in Solved Problem #a to find sec 45. Use the definition of the cosecant function. hypotenuse csc 45 opposite 1 b. Use the right triangle to find tan 60. b. Use the right triangle in Solved Problem #b to find cos0. Use the definition of the tangent function and the angle marked 60 in the triangle. opposite tan 60 adjacent 1 Copyright 014 Pearson Education Inc. 17

12 Algebra and Trigonometry 5e Objective #: Recognize and use fundamental identities. Solved Problem # Pencil Problem # 5 a. Given sinθ and cos θ, find the value of each of the four remaining trigonometric functions. Find tanθ using a quotient identity. 1 a. Given sinθ and cos θ, find the value of each of the four remaining trigonometric functions. sinθ 5 5 tanθ cosθ Use reciprocal identities to find the remaining three function values. 1 1 cscθ sinθ secθ cosθ cotθ tanθ 5 1 b. Given that sinθ and θ is an acute angle, find the value of cosθ using a trigonometric identity. 6 b. Given that sinθ and θ is an acute angle, find 7 the value of cosθ using a trigonometric identity. Use the Pythagorean identity sin θ cos θ 1. Because θ is an acute angle, cosθ is positive. 1 cos θ 1 1 cos θ cos θ cosθ Copyright 014 Pearson Education Inc.

13 Objective #4: Use cofunctions of complements. Solved Problem #4 4a. Find a cofunction with the same value as sin 46. sin 46cos(9046 ) cos 44 Pencil Problem #4 Section 5. 4a. Find a cofunction with the same value as sin 7. 4b. Find a cofunction with the same value as cot. 1 4b. Find a cofunction with the same value as tan cot tan tan tan Objective #5: Evaluate trigonometric functions with a calculator. Solved Problem #5 5a. Use a calculator to find the value of sin 7.8 to four decimal places. Use degree mode. On a scientific calculator, enter the angle measure, 7.8, and then press the SIN key. On a graphing calculator, press the SIN key, and then enter the angle measure, 7.8, and press ENTER. The display, rounded to four places, should be Pencil Problem #5 5a. Use a calculator to find the value of tan.7 to four decimal places. 5b. Use a calculator to find the value of csc1.5 to four decimal places. Use radian mode. 5b. Use a calculator to find the value of cot to four 1 decimal places. On a scientific calculator, enter the angle measure, 1.5, and then press the SIN key followed by the reciprocal key labeled 1 x. On a graphing calculator, open a set of parentheses, press the SIN key, and then enter the angle measure, 1.5. Close the parentheses, press the reciprocal key labeled x 1, and press ENTER. The display, rounded to four places, should be Copyright 014 Pearson Education Inc. 175

14 Algebra and Trigonometry 5e Objective #6: Use right triangle trigonometry to solve applied problems. Solved Problem #6 6. The distance across a lake, a, is unknown. To find this distance, a surveyor took the measurements shown in the figure. What is the distance across the lake? Pencil Problem #6 6. To find the distance across a lake, a surveyor took the measurements shown in the figure. Use the measurements to determine how far it is across the lake. Round to the nearest yard. We know the measurements of one angle and the leg adjacent to the angle. We need to know the length of the side opposite the known angle. We use the tangent function. a tan a 750 tan 4 a.9 The distance across the lake is approximately.9 yards. Answers for Pencil Problems (Textbook Exercise references in parentheses): 1a. 1b sin θ, cos θ, tan θ, csc θ, sec θ, cotθ (5. #1) sin θ, cos θ, tan θ, csc θ, sec θ, cotθ (5. #) a. (5. #11) b. (5. #9) a. tan θ, cscθ, sec θ, cotθ (5. #19) b a. cos8 (5. #1) 4b. cot (5. #5) 18 5a (5. #41) 5b..71 (5. #47) yd (5. #7) 176 Copyright 014 Pearson Education Inc. 1 7 (5. #1)

15 Section 5. Trigonometric Functions of Any Angle Could you repeat that? There are many repetitive patterns in nature. Tides cycle through a pattern of low and high tides in a very predictable manner. The number of hours of daylight on a given day varies throughout the year, but the pattern throughout the year repeats itself year after year. In this section, we extend the definitions of the trigonometric functions to include all angles. In doing so, we begin to see the repetitive properties of the trigonometric functions that make them useful for modeling cyclic phenomena. Objective #1: Use the definitions of the trigonometric functions of any angle. Solved Problem #1 1a. Let P = (1, ) be a point on the terminal side of θ. Find each of the six trigonometric functions of θ. We are given that x = 1 and y =. We need the value of r. Pencil Problem #1 1a. Let P = (, 5) be a point on the terminal side of θ. Find each of the six trigonometric functions of θ. r x y 1 ( ) Now we use the definitions of the trigonometric functions of any angle. y sinθ r x cosθ r y tanθ x 1 r cscθ y r 10 secθ 10 x 1 x 1 1 cotθ y Copyright 014 Pearson Education Inc. 177

16 Algebra and Trigonometry 5e 1b. Evaluate, if possible: csc180. The terminal side of θ = 180 is on the negative x-axis. We select the point ( 1, 0) on the terminal side of the angle, which is 1 unit from the origin, so x = 1, y = 0, and r = 1. r 1 cscθ y 0 csc180 is undefined. 1b. Evaluate, if possible: tan. Objective #: Use the signs of the trigonometric functions. Solved Problem # Pencil Problem # a. If sinθ 0 and cosθ 0, name the quadrant in a. If tanθ 0 and cosθ 0, name the quadrant in which θ lies. which θ lies. When sinθ 0, θ lies in quadrant III or IV. When cosθ 0, θ lies in quadrant II or III. When both conditions are met, θ must lie in quadrant III. b. Given that and sec θ. 1 tanθ and cosθ 0, find sinθ b. Given that and csc θ. tanθ and sinθ 0, find cosθ Because both the tangent and cosine are negative, θ lies in quadrant II, where x is negative and y is positive. 1 y 1 tanθ x So, x = and y = 1. Find r. r x y ( ) Now use the definitions of the trigonometric functions of any angle. y sinθ r r secθ x 178 Copyright 014 Pearson Education Inc.

17 Section 5. Solved Problem # a. Find the reference angle for θ = 10. The angle lies in quadrant III. The reference angle is θ Objective #: Find reference angles. Pencil Problem # a. Find the reference angle for θ = 160. b. Find the reference angle for θ = 7. 4 b. Find the reference angle for θ = 5. 6 The angle lies in quadrant IV. The reference angle is θ c. Find the reference angle for θ = 40. c. Find the reference angle for θ = 5. The angle lies in quadrant II. The positive acute angle formed by the terminal side of θ and the x-axis is 60. The reference angle isθ 60. d. Find the reference angle for θ = 665. d. Find the reference angle for θ = 565. Subtract 60 to find a positive coterminal angle less than 60 : = 05. The angle α = 05 lies in quadrant IV. The reference angle is α Copyright 014 Pearson Education Inc. 179

18 Algebra and Trigonometry 5e e. Find the reference angle for θ = 11. e. Find the reference angle for θ = Add 4 to find a positive coterminal angle less than : 4. The angle α lies in quadrant I. The reference angle is α. Objective #4: Use reference angles to evaluate trigonometric functions. Solved Problem #4 4a. Use a reference angle to find the exact value of sin 00. A 00 angle lies in quadrant IV, where the sine function is negative. The reference angle is θ Pencil Problem #4 4a. Use a reference angle to find the exact value of cos5. sin 00sin 60 4b. Use a reference angle to find the exact value of 5 tan. 4 4b. Use a reference angle to find the exact value of sin. A 5 angle lies in quadrant III, where the tangent 4 function is positive. The reference angle is θ tan tan Copyright 014 Pearson Education Inc.

19 Section 5. 4c. Use a reference angle to find the exact value of sec. 6 4c. Use a reference angle to find the exact value of tan. 4 A angle lies in quadrant IV, where the secant 6 function is positive. Furthermore, a angle 6 forms an acute of with the x-axis. The reference 6 angle is θ. 6 sec sec 6 6 4d. Use a reference angle to find the exact value of 17 cos. 6 4d. Use a reference angle to find the exact value of 19 cot. 6 Subtract to find a positive coterminal angle less than : A 5 angle lies in quadrant II, where the cosine 6 function is negative. The reference angle is θ cos cos 6 6 Copyright 014 Pearson Education Inc. 181

20 Algebra and Trigonometry 5e Answers for Pencil Problems (Textbook Exercise references in parentheses): a. sin θ, cos θ, tan θ, csc θ, sec θ, cotθ (5. #7) b. undefined (5. #1) a. quadrant II (5. #1) b. a. 0 (5. #5) b. 4a. 6 (5. #61) 4b. 1 1 cos θ, cscθ (5. #9) 1 (5. #4) c. 5 (5. #47) d. 5 (5. #51) e. (5. #67) 4c. 1 (5. 75) 4d. (5. #79) 4 (5. #57) 18 Copyright 014 Pearson Education Inc.

21 Section 5.4 Trigonometric Functions of Real Numbers; Periodic Functions Could you repeat that one more time?? In the last section, we moved from defining the trigonometric functions for only acute angles in right triangles to defining them for all angles in standard position. We used properties of the Cartesian coordinate system to help us evaluate the trigonometric functions more efficiently and we began to see repetitive patterns in the function values. In this section, we define the trigonometric functions for real numbers instead of angles. By defining the functions in terms of movement around a circle, we see that the functions values will begin to repeat themselves when we complete one full trip around the circle and begin a second. In doing so, we make the repetitive properties of the trigonometric functions more apparent. Objective #1: Use a unit circle to define trigonometric functions of real numbers. Solved Problem #1 1. Use the figure to find the values of the trigonometric functions at t. Pencil Problem #1 1. Use the figure to find the values of the trigonometric functions at t. The point on the circle has coordinates 1,. (continued on next page) Copyright 014 Pearson Education Inc. 18

22 Algebra and Trigonometry 5e We use x = and 1 y = in the definitions. 1 sin t = y = cost = x = 1 y 1 1 tant = = = = = x 1 1 csct = y = 1 = 1 1 sect = = = = = x x cot t = = y 1 = Objective #: Recognize the domain and range of sine and cosine functions. Solved Problem #. True or false: The range of the cosine function is all real numbers, (, ), so there is a real number t in the domain of the cosine function for which cost = 10. False; the range of cosine is [ 1, 1] and 10 is not in this interval, so there is no real number t in the domain of the cosine function for which cost = 10. Pencil Problem #. True or false: There is a real number t in the domain 10 of the sine function for which sin t =. 184 Copyright 014 Pearson Education Inc.

23 Objective #: Use even and odd trigonometric functions. Solved Problem # a. Use even and odd properties to find the exact value of cos( 60 ). The cosine function is even: cos( t) = cos t. 1 cos( 60 ) = cos60 = Pencil Problem # Section 5.4 a. Use even and odd properties to find the exact value of cos. 6 b. Use even and odd properties to find the exact value of tan. 6 b. Use even and odd properties to find the exact value 5 of sin. 6 The tangent function is odd: tan( t) = tan t. tan = tan = 6 6 Copyright 014 Pearson Education Inc. 185

24 Algebra and Trigonometry 5e Solved Problem #4 4a. Use periodic properties to find the exact value of cos 405. The period of cosine is or 60 : cos( t+ 60 ) = cos t. Objective #4: Use periodic properties. Pencil Problem #4 4a. Use periodic properties to find the exact value of cos810. cos 405 = cos( ) = cos 45 = 4b. Use periodic properties to find the exact value of 7 sin. 4b. Use periodic properties to find the exact value of 11 sin. 4 The period of sine is : sin( t+ ) = sin t. 7 6 sin = sin + = sin + = sin = Answers for Pencil Problems (Textbook Exercise references in parentheses): sin t =, cos t =, tan t =, csc t =, sec t =, cot t = (5.4 #1) false (5.4 #54) a. (5.4 #19b) b. 4a. 0 (5.4 #7b) 4b. 1 (5.4 #1b) (5.4 #5b) 186 Copyright 014 Pearson Education Inc.

25 Section 5.5 Graphs of Sine and Cosine Functions The Graph That Looks Like a Rollercoaster In this section, we study the graphs of the sine and cosine functions. The ups and downs of the graphs may remind you of a rollercoaster. The graphs rise to peaks at maximum points then plunge to minimum points where they promptly change direction and rise again. However, unlike a rollercoaster ride, the ups and downs of these graphs go on forever. The shapes of these graphs and their repetitive properties make it even more obvious why trigonometric functions are used to model cyclic behavior. Objective #1: Understand the graph of y= sin x. Solved Problem #1 1. True or false: The graph of y sin x is symmetric with respect to the y-axis. Pencil Problem #1 1. True or false: The graph of y sin x has no gaps or holes and extends indefinitely in both directions. False; the sine function is an odd function and is symmetric with respect to the origin but not the y-axis. Objective #: Graph variations of y= sin x. Solved Problem # a. Determine the amplitude and the period of 1 y sin x. Then graph the function for 0 x 8. Pencil Problem # a. Determine the amplitude and the period of 1 y = sin x. Then graph one period of the function. Comparing 1 y sin x to y = Asin Bx, we see that A = and B = 1. Amplitude: A = = Period: = = 4 B 1 The amplitude tells us that the maximum value of the function is and the minimum value is. The period tells us that the graph completes one cycle between 0 and 4. (continued on next page) Copyright 014 Pearson Education Inc. 187

26 Algebra and Trigonometry 5e Divide the period by 4: 4 =. The x-values of the 4 five key points are x 1 = 0, x = 0 + =, x = + =, x4 = + =, and x5 = + = 4. Now find the value of y for each of these x-values. 1 y = sin 0= sin0= 0= 0:(0, 0) 1 y = sin = sin = 1= :(, ) 1 y sin = = sin = 0 = 0:(, 0) 1 y = sin = sin = ( 1) = :(, ) 1 y sin = 4 = sin = 0 = 0:(4, 0) Notice the pattern: x-intercept, maximum, x-intercept, minimum, x-intercept. Plot these points and connect them with a smooth curve. Extend the graph one period to the right in order to graph from 0 x Copyright 014 Pearson Education Inc.

27 Section 5.5 b. Determine the amplitude, period, and phase shift of y = sin x. Then graph one period of the function. b. Determine the amplitude, period, and phase shift of y= sin( x ). Then graph one period of the function. Comparing y = sinx to y = Asin( Bx C), we see that A =, B =, and C =. Amplitude: A = = Period: = = B Phase shift: C B 1 = = = 6 The amplitude tells us that the maximum value of the function is and the minimum value is. The period tells us that each cycle is of length. The phase shift tells us that a cycle starts at. 6 Divide the period by 4:. The x-values of the five 4 key points are x1 =, 6 5 x = + = + =, x = + = + = =, x4 = + = + =, and x5 = + = + = =. Now find the value of y for each of these x-values. y = sin = sin0= 0:, y = sin = sin = :, 1 1 y = sin = sin = 0:, y = sin = sin = :, y = sin = sin = 0:, (continued on next page) Copyright 014 Pearson Education Inc. 189

28 Algebra and Trigonometry 5e Notice the pattern: x-intercept, maximum, x-intercept, minimum, x-intercept. Plot these points and connect them with a smooth curve. Objective #: Understand the graph of y= cos x.. Solved Problem #. True or false: The graph of y = cos x is symmetric with respect to the y-axis. Pencil Problem #. True or false: The graph of y = cos x illustrates that the range of the cosine function is (, ). True; the cosine function is an even function, and the graphs of even functions have symmetry with respect to the y-axis. Objective #4: Graph variations of y= cos x. Solved Problem #4 4. Determine the amplitude, period, and phase shift of y = cos( x+ ). Then graph one period of the function. Pencil Problem #4 4. Determine the amplitude, period, and phase shift of 1 y = cos x+. Then graph one period of the function. Comparing y = cos( x+ ) to y = Asin( Bx C), we see that A =, B =, and C =. Amplitude: A = = Period: = = B C Phase shift: = = B (continued on next page) 190 Copyright 014 Pearson Education Inc.

29 Section 5.5 The amplitude tells us that the maximum value of the function is and the minimum value is. The period tells us that each cycle is of length. The phase shift tells us that a cycle starts at. Divide the period by 4:. The x-values of the five 4 key points are x1 =, x = + = + =, x = + = 0, x4 = 0 + =, and x5 = + = =. Now find the value of y for each of these x-values by evaluating the function. The five key points are,,, 0, 0,,, 0, 4 4 and,. Notice the pattern: maximum, x-intercept, minimum, x-intercept, maximum. Plot these points and connect them with a smooth curve. Copyright 014 Pearson Education Inc. 191

30 Algebra and Trigonometry 5e Objective #5: Use vertical shifts of sine and cosine curves. Solved Problem #5 5. Graph one period of the function y = cosx+ 1. The graph of y = cosx+ 1 is the graph of y = cosx shifted one unit up. The amplitude is and the period is. The graph oscillates units below and units above the line y = 1. Pencil Problem #5 5. Graph one period of the function y = sin x+. The x-values for the five key points are 0,,,, and. The five key points are (0, ),, 1, (, 1),, 1, and (, ). Plot these points and connect them with a smooth curve. 19 Copyright 014 Pearson Education Inc.

31 Solved Problem #6 6. A region that is 0 north of the Equator averages a minimum of 10 hours of daylight in December. Hours of daylight are at a maximum of 14 hours in June. Let x represent the month of the year, with 1 for January, for February, for March, and 1 for December. If y represents the number of hours of daylight in month x, use a sine function of the form y = Asin( Bx C) + D to model the hours of daylight. We need to determine values for A, B, C, and D in y= Asin( Bx C) + D. To find D, notice that the values range between a minimum of 10 and a maximum of 14. The middle value is 1, so D = 1. To find A, notice that the minimum, 10, and maximum, 14, are units below and above the middle value, D = 1. The amplitude is, so A =. To find B, notice that the period is 1 months (one year). Use the period formula and solve for B. = 1 B = 1B B = = 1 6 To find C, notice that a middle value occurs in March (x = ), so we can begin a cycle in March. Use the phase shift formula with this value of x and the value of B just found. Bx C = 0 C = 0 6 = C Substitute the values for A, B, C, and D into y= Asin( Bx C) + D. The model is y = sin x Objective #6: Model periodic behavior. Pencil Problem #6 Section The figure shows the depth of water at the end of a boat dock. The depth is 6 feet at low tide and 1 feet at high tide. On a certain day, low tide occurs at 6 a.m. and high tide occurs at noon. If y represents the depth of the water x hours after midnight, use a cosine function of the form y = Acos Bx+ D to model the water s depth. Copyright 014 Pearson Education Inc. 19

32 Algebra and Trigonometry 5e Answers for Pencil Problems (Textbook Exercise references in parentheses): 1. true a. Amplitude: ; period: 4 (5.5 #9) b. Amplitude: ; period: ; phase shift:. false (5.5 #1) 4a. Amplitude: 1 ; period: ; phase shift: (5.5 #47) 6 5. (5.5 #5) x 6. y = cos + 9 (5.5 #87) Copyright 014 Pearson Education Inc.

33 Section 5.6 Graphs of Other Trigonometric Functions Trig Functions Without Bounds We have seen that sine and cosine functions can be used to model phenomena that are cyclic in nature, such as the number of daylight hours in a day at a specific location over a period of years. But it s not just the periodic properties of sine and cosine that are important in these situations; it s also their limited ranges. The fact that the ranges of sine and cosine functions are bounded makes them suitable for modeling phenomena that never exceed certain minimum and maximum values. In this section, we will see that the remaining trigonometric functions, while periodic like sine and cosine, do not have bounded ranges like sine and cosine. These functions are more suitable for describing cyclic phenomena that do not have natural limits, such as the location where a beam of light from a rotating source strikes a flat surface. Objective #1: Understand the graph of y tan x. Solved Problem #1 1. True or false: The graph of y tan x is symmetric with respect to the y-axis. Pencil Problem #1 1. True or false: The graph of y tan x has no gaps or holes and extends indefinitely in both directions. False; the tangent function is an odd function and is symmetric with respect to the origin but not the y-axis. Objective #: Graph variations of y tan x. Solved Problem # a. Graph y tanx for x. 4 4 Find two consecutive asymptotes by solving Bx C. In this case, Bx C x. Pencil Problem # 1 a. Graph two full periods of y tan x. x x 4 4 (continued on next page) Copyright 014 Pearson Education Inc. 195

34 Algebra and Trigonometry 5e The graph completes one cycle on the interval, and has consecutive vertical asymptotes 4 4 at x and x. The x-intercept is midway 4 4 between the asymptotes at (0, 0). The x-values 8 1 and are 8 4 and of the way between the 4 asymptotes; the points, 8 and 8, are on the graph. Plot these three points and the asymptotes. Graph one period of the function by drawing a smooth curve through the points and approaching the asymptotes. Complete one more period to the right in order to show the graph for x. 4 4 b. Graph two full periods of y tan x. b. Graph two full periods of y x tan. Find two consecutive asymptotes by solving Bx C. In this case, Bx C x. x x 0 x (continued on next page) 196 Copyright 014 Pearson Education Inc.

35 Section 5.6 The graph completes one cycle on the interval (0, ) and has consecutive vertical asymptotes at x = 0 and x =. The x-intercept is midway between the asymptotes at, 0. The x-values and 4 4 are 1 4 and 4 of the way between the asymptotes; the points, 1 4 and, 1 4 graph. are on the Plot these three points and the asymptotes. Graph one period of the function by drawing a smooth curve through the points and approaching the asymptotes. Complete one more period to the left or to the right in order to show two full periods. The graph of y tan x is the graph of y tan x shifted units to the right. Objective #: Understand the graph of y cot x. Solved Problem #. True or false: The graph of y cot x has asymptotes at values of x where the graph of y tan xhas x-intercepts. Pencil Problem #. True or false: The graph of y cot x illustrates that the range of the cotangent function is (, ). True; the cotangent function is undefined at values of x for which tangent is 0. Copyright 014 Pearson Education Inc. 197

36 Algebra and Trigonometry 5e Objective #4: Graph variations of y cot x. 4. Graph Solved Problem #4 1 y cot x. Pencil Problem # Graph two periods of y cot x. Find two consecutive asymptotes by solving 0 Bx C. In this case, Bx C x. 0 x 0 x 0 x The graph completes one cycle on the interval (0, ) and has consecutive vertical asymptotes at x 0 and x. The x-intercept is midway between the asymptotes at (1, 0). The x-values 1 and are 1 4 and 4 of the way between the asymptotes; the 1 1 points, and 1, are on the graph. Plot these three points and the asymptotes. Graph one period of the function by drawing a smooth curve through the points and approaching the asymptotes. 198 Copyright 014 Pearson Education Inc.

37 Section 5.6 Objective #5: Understand the graphs of Solved Problem #5 y csc x and y sec x. Pencil Problem #5 5. True or false: The graph of y csc x illustrates that the range of the cosecant function is (, ). False; the range of the cosecant function is (, 1] [1, ). It does not include values of y between 1 and True or false: The graph of y sec x has asymptotes at values of x where the graph of y sin x has x-intercepts. Objective #6: Graph variations of Solved Problem #6 y csc x and y sec x. Pencil Problem #6 6. Graph y secx for x. 4 4 Begin by graphing y cosx where secant has been replaced by cosine, its reciprocal function. The amplitude of y cosx is and the period is. Note that for x the graph of 4 4 y cosx has x-intercepts at x, x, x, and x. The graph of y cosx attains its minimum value of at x and x and its maximum value of at x = Graph two periods of 1 x y csc. At each x-intercept of the graph of y cosx draw a vertical asymptote. Graph y secxby starting at each minimum or maximum point and approaching the asymptotes in each direction. Copyright 014 Pearson Education Inc. 199

38 Algebra and Trigonometry 5e Answers for Pencil Problems (Textbook Exercise references in parentheses): 1. false a. (5.6 #7) b. (5.6 #11). true 4. (5.6 #19) 5. false 6.. (5.6 #1) 00 Copyright 014 Pearson Education Inc.

39 Section 5.7 Inverse Trigonometric Functions For Your Viewing Pleasure Your total movie experience at your local cinema is affected by many variables, including your distance from the screen. If you sit too close, your viewing angle is too small. If you sit too far back, the image is too small. In this section s Exercise Set, you will see how inverse trigonometric functions can be used to model your viewing angle in terms of your distance from the screen and help you find the seat that will optimize your viewing pleasure. Objective #1: Understand and use the inverse sine function. Solved Problem #1 1 1a. Find the exact value of sin. 1 Let θ sin. Then sinθ where θ. We must find the angle θ, θ, whose sine equals. Using a table of values for the sine function for θ, we see that sin. Pencil Problem # a. Find the exact value of sin. Thus, 1 sin. Copyright 014 Pearson Education Inc. 01

40 Algebra and Trigonometry 5e 1b. Find the exact value of 1 sin. 1b. Find the exact value of sin Let θ sin. Then sinθ where θ. We must find the angle θ, θ, whose sine equals. Using a table of values for the sine function for θ, we see that sin. 4 1 Thus, sin. 4 Objective #: Understand and use the inverse cosine function Solved Problem # Pencil Problem #. Find the exact value of cos. Let θ cos Find the exact value of cos. Then 1 cosθ where 0 θ. We must find the angle θ, 0 θ, whose cosine 1 equals. Using a table of values for the cosine 1 function for 0 θ, we see that cos. Thus, 1 1 cos. 0 Copyright 014 Pearson Education Inc.

41 Section 5.7. Find the exact value of 1 Let θ tan ( 1). Objective #: Understand and use the inverse tangent function. Solved Problem # 1 tan ( 1). Then tanθ 1 where θ. We must find the angle θ, θ, whose tangent equals 1. Using a table of values for the tangent function for θ, we see that tan Find the exact value of Pencil Problem # 1 tan 0. Thus, 1 tan ( 1). 4 Objective #4: Use a calculator to evaluate inverse trigonometric functions. Solved Problem #4 Pencil Problem #4 4a. Use a calculator to find the value of cos decimal places. 1 1 to four 4a. Use a calculator to find the value of cos decimal places. 1 8 to two To access the inverse cosine function, you will need to press the nd function key and then the COS key. Use radian mode. Scientific calculator: 1 = nd COS Graphing calculator: nd COS (1 ) ENTER The display should read 1.10, rounded to four places. Copyright 014 Pearson Education Inc. 0

42 Algebra and Trigonometry 5e 4b. Use a calculator to find the value of to four decimal places. 1 tan ( 5.85) 4b. Use a calculator to find the value of to two decimal places. 1 sin ( 0.) To access the inverse tangent function, you will need to press the nd function key and then the TAN key. Use radian mode. Scientific calculator: / nd TAN Graphing calculator: nd TAN ( ) 5.85 ENTER The display should read 1.549, rounded to four places. Objective #5: Find exact values of composite functions with inverse trigonometric functions. Solved Problem #5 Pencil Problem #5 5a. Find the exact value, if possible: 1 cos(cos 0.7). 5a. Find the exact value, if possible: 1 sin(sin 0.9). Since 0.7 is in the interval [ 1, 1], we can use the 1 inverse property cos(cos x) = x. 1 cos(cos 0.7) = b. Find the exact value, if possible: sin (sin ). Since is not in the,, we cannot use the 1 inverse property sin (sin x) = x. We first evaluate sin = 0, and then evaluate 1 sin b. Find the exact value, if possible: sin sin sin (sin ) = sin 0 = 0 04 Copyright 014 Pearson Education Inc.

43 Section 5.7 5c. Find the exact value, if possible: 1 cos[cos ( 1.)]. 5c. Find the exact value, if possible: 1 sin(sin ). Since 1. is not in the domain of the inverse cosine 1 function, [ 1, 1], cos[cos ( 1.)] is not defined. 5d. Find the exact value of sintan. 4 1 Let θ represent the angle in, whose tangent is. 4 Thus, 1 θ = tan. 4 So, tan θ =, where < θ <. Since tanθ is 4 positive, θ must be in 0,. Thus, θ lies in quadrant I, where both x and y are positive. 5d. Find the exact value of cossin y tan θ = =, so x = 4 and y = 4 x Find r and then use the value of r to find sin θ. r = 4 + = = 5 = 5 y sinθ = = r 5 Thus, 1 sin tan = sin θ =. 4 5 Copyright 014 Pearson Education Inc. 05

44 Algebra and Trigonometry 5e 5e. Find the exact value of 1 1 cos sin. 5e. Find the exact value of 1 csc cos. Let θ represent the angle in, Thus, θ = sin. whose sine is 1 So, sin θ =, where θ. Since sinθ is negative, θ must be in, 0. Thus, θ lies in quadrant IV, where both x is positive and y is negative. 1 y 1 sin θ = = =, so y = 1 and r = r r Find x and then use the value of x to find cos θ. r = x + y = x + ( 1) 4= x + 1 = x x=, since x> 0 x cosθ = = r Thus, 1 1 cos sin = cos θ =. In this problem, we know how to find the exact 1 1 value of sin, so we could have also proceeded as follows: 1 1 cos sin = cos = Copyright 014 Pearson Education Inc.

45 Section f. If x > 0, write sec(tan x) expression in x. as an algebraic 1 5f. If x > 0, write tan(cos x) expression in x. as an algebraic Let θ represent the angle in, whose tangent is x. 1 Thus, θ = tan x and tan θ = x, where,. Because x > 0, tanθ is positive. Thus, θ is a firstquadrant angle. Draw a right triangle and label one of the acute angles θ. Since x opposite tan θ = x = =, 1 adjacent the length of the side opposite θ is x and the length of the adjacent side is 1. Use the Pythagorean theorem to find the hypotenuse. c c = x + 1 = x + 1 hypotenuse x + 1 Thus, secθ = = = x + 1 and adjacent 1 consequently, 1 sec(tan x) = secθ = x + 1. Copyright 014 Pearson Education Inc. 07

46 Algebra and Trigonometry 5e Answers for Pencil Problems (Textbook Exercise references in parentheses): 1a. 6 (5.7 #1) 1b. (5.7 #5) 6. 4 (5.7 #9). 0 (5.7 #15) 4a (5.7 #) 4b. 0. (5.7 #1) 5a. 0.9 (5.7 #1) 5b. 5e. (5.7 #59) 5f. 6 1 x x (5.7 #5) 5c. undefined (5.7 #45) 5d. (5.7 #6) 5 (5.7 #47) 08 Copyright 014 Pearson Education Inc.

47 Section 5.8 Applications of Trigonometric Functions Up, Up, & Away, Around the Corner, and Back & Forth From finding the heights of tall buildings to modeling cyclic behavior, trigonometry is very useful. In this section, you will see how finding missing parts of right triangles has many practical applications, how ships at sea can be located using bearings and trigonometry, and even how the motion of a ball attached to a spring can be described using a sinusoidal function. Solved Problem #1 1a. Let A = 6.7 and a = 8.4 in the triangle shown below. Solve the right triangle, rounding lengths to two decimal places. Objective #1: Solve a right triangle. Pencil Problem #1 1a. Let A =.5 and b = 10 in the triangle shown in Solved Problem #1a. Solve the right triangle, rounding lengths to two decimal places. To solve the triangle, we need to find B, b, and c. We begin with B. We know that AB B 90 B Next we find b. Note that b is opposite B = 7. and a = 8.4 is adjacent to B = 7.. We set up an equation relating a, b, and B using the tangent function. Then we solve for b. side opposite B b tan B side adjacent to B a b tan b 8.4 tan (continued on next page) Copyright 014 Pearson Education Inc. 09

48 Algebra and Trigonometry 5e When finding c, we choose not to use the value of b just found because it was rounded. We will again use B = 7. and a = 8.4. Since a is adjacent to B and we are finding the hypotenuse, we use the cosine function. side adjacent to B a cos B hypotenuse c 8.4 cos7. c c cos c 9.45 cos7. Thus, B = 7., b 4.4, and c Note that angle A could have been used in the a calculations to find b and c, tan A and b a sin A, without using rounded values. c 1b. From a point on level ground 80 feet from the base of a tower, the angle of elevation is Approximate the height of the tower to the nearest foot. 1b. From a point on level ground 580 feet (1 mile) from the base of a television transmitting tower, the angle of elevation is 1.. Approximate the height of the tower to the nearest foot. We draw a right triangle to illustrate the situation. The tower and the ground form the right angle. One of the acute angles measures 85.4, and the side adjacent to it is 80 feet. The tower is opposite the 85.4 angle; we let h represent the height of the tower. (continued on next page) 10 Copyright 014 Pearson Education Inc.

49 Section 5.8 Since we are looking for the side opposite the known acute angle and we know the side adjacent to that angle, we will use tangent. opposite side h tan85.4 adjacent side 80 h 80 tan The tower is approximately 994 feet tall. 1c. A guy wire is 1.8 yards long and is attached from the ground to a pole at a point 6.7 yards above the ground. Find the angle, to the nearest tenth, that the wire makes with the ground. 1c. A wheelchair ramp is built beside the steps to the campus library. Find the angle of elevation of the -foot ramp, to the nearest tenth of a degree, if its final height is 6 feet. We draw a right triangle to illustrate the situation. The pole and the ground form the right angle. We are looking for the angle opposite the pole; call it A. The side opposite a is 6.7 yards, and the hypotenuse is 1.8 yards. Since we know the side opposite A and the hypotenuse, we will use the sine function. side opposite A 6.7 sin A hypotenuse A sin The wire makes an angle of approximately 9.0 with the ground. Copyright 014 Pearson Education Inc. 11

50 Algebra and Trigonometry 5e 1d. 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, and the angle of elevation to the top is 5. Find the height of the sculpture of Lincoln s face to the nearest tenth of a foot. 1d. A hot-air balloon is rising vertically. From a point on level ground 15 feet from the point directly under the passenger compartment, the angle of elevation changes from 19. to 1.7. How far, to the nearest tenth of a foot, does the balloon rise during this period? Refer to the figure below, where a represents the distance to the bottom of the sculpture and b represents the distance to the top of the sculpture. The height of the sculpture is then b a. We need to find a and b and then subtract. Since in each case we are finding the length of the leg opposite a known angle and we also know the length of the leg adjacent to the known angle, we will use the tangent function twice. opposite side a tan adjacent side 800 a 800 tan opposite side b tan 5 adjacent side 800 b 800 tan ba Lincoln s face is approximately 60. feet tall. 1 Copyright 014 Pearson Education Inc.

51 Objective #: Solve problems involving bearings. Solved Problem #. You hike. miles on a bearing of S 1 W. Then you turn 90 clockwise and hike.5 miles on a bearing of N 59 W. At that time, what is your bearing, to the nearest tenth of a degree, from your starting point? The figure below illustrates the situation. Notice that we have formed a triangle by drawing a segment from the starting point to the ending point. The triangle is a right triangle because of the 90 change in direction. We know the lengths of the legs of the right triangle. Pencil Problem # Section 5.8. After takeoff, a jet flies 5 miles on a bearing of N 5 E. Then it turns 90 and flies 7 miles on a bearing of S 55 E. At that time, what is the bearing of the jet, to the nearest tenth of a degree, from the point where it took off? We labeled one of the acute angles θ. The side opposite θ is.5 miles and the side adjacent to θ is. miles. We can use the tangent function to find θ. opposite side.5 tanθ adjacent side. 1.5 θ tan Now the angle formed by the north-south line through the starting point and the segment between the starting and ending points is θ = The bearing of the ending point from the starting point is S 86.7 W. Copyright 014 Pearson Education Inc. 1

52 Algebra and Trigonometry 5e Objective #: Model simple harmonic motion. Solved Problem # a. A ball on a string 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. At t = 0 seconds, d = 6 inches. We use a negative value for d because the motion begins with the ball below its rest position. Also, because the motion begins when the ball is at its greatest distance from rest, we use the model containing cosine rather than the one containing sine. We need to find values for a and ω in d acos ωt. Since the maximum displacement is 6 inches and the ball is initially below its rest position, a = 6. The period is given as 4 seconds. Use the period formula to find ω. 4 ω 4ω ω 4 The equation is d 6cos t. Pencil Problem # a. A ball on a string is pulled 8 inches below its rest position and then released. The period for the motion is seconds. Write the equation for the ball s simple harmonic motion. 14 Copyright 014 Pearson Education Inc.

53 Section 5.8 b. An object moves in simple harmonic motion described by d 1cos t, where t is measured in 4 seconds and d in centimeters. Find the maximum displacement, the frequency, and the time required for one cycle. b. An object moves in simple harmonic motion described by d 5cos t, where t is measured in seconds and d in inches. Find the maximum displacement, the frequency, and the time required for one cycle. The maximum displacement is the amplitude. Because a = 1, the maximum displacement is 1 centimeters. The frequency, f, is f ω The frequency is 1 oscillation per second. 8 The time required for one cycle is the period. period ω The time required for one cycle is 8 seconds. Copyright 014 Pearson Education Inc. 15

54 Algebra and Trigonometry 5e Answers for Pencil Problems (Textbook Exercise references in parentheses): 1a. B = 66.5, a 4.5, c (5.8 #1) 1b. 059 feet (5.8 #41) 1c (5.8 #47) 1d..7 feet (5.8 #49). N 89.5 E (5.8 #57) a. d 8cost (5.8 #18) b. maximum displacement: 5 inches; frequency: 1 4 (5.8 #1) oscillation per second; time for one cycle: 4 seconds 16 Copyright 014 Pearson Education Inc.