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CC-31 Reciprocal Trigonometric Functions Common Core State Standards MACC.912.F-IF.3.7e Graph . . . trigonometric functions, showing period, midline, and amplitude. MP 1, MP 2, MP 3, MP 4, MP 5 Objectives To evaluate reciprocal trigonometric functions To graph reciprocal trigonometric functions This asks only for the length of the extension, not the length of the extension ladder. You want the extension ladder to reach the window sill so you can wash the top window. What expression gives the length by which you should extend the ladder while keeping the base in place? Explain. 5 ft 20 ft MATHEMATICAL PRACTICES Lesson Vocabulary •cosecant •secant •cotangent 2Hon_SE_CC_31_TrKit.indd 151 /13 7:19 PM 70 To solve an equation ax = b, you multiply each side by the reciprocal of a. If a is a trigonometric expression, you need to use its reciprocal. Essential Understanding Cosine, sine, and tangent have reciprocals. Cosine and secant are reciprocals, as are sine and cosecant. Tangent and cotangent are also reciprocals. Key Concept Cosecant, Secant, and Cotangent Functions The cosecant (csc), secant (sec), and cotangent (cot) functions are defined using reciprocals. Their domains do not include the real numbers u that make the denominator zero. 1 1 sec u = cos csc u = sin u u (cot u = 0 at odd multiples of p2 , where tan u is undefined.) 1 cot u = tan u You can use the unit circle to evaluate the reciprocal trigonometric functions directly. Suppose the terminal side of an angle u in standard position intersects the unit circle at the point (x, y). Then csc u = 1 y , sec u = 1 x , cot u = x y. Lesson 13-8 Reciprocal Trigonometric Functions CC-31 Reciprocal Trigonometric Functions y P (x, y) u O 1 x 151 8/5/13 151 7:20 PM You can use what you know about the unit circle to find exact values for reciprocal trigonometric functions. Problem 1 Finding Values Geometrically ( ) ( ) P What are the exact values of cot − 5P 6 and csc 6 ? Do not use a calculator. Find the point where the unit circle intersects the terminal side of the angle - 5p 6 radians. ( 5p6 ) cot - y x (– √23 , – 21( Find the exact value of cot - 5p 6 . ( ) ( cot − –5p 6 5p x =y 6 ) 23 − 2 = ( cot − Find the point where the unit circle intersects the terminal side of the angle p 6 radians. Find the exact value of csc p6 . ( ) csc − 21 5p = 13 6 ) ( p6 ) y (√32 , 12 ) x csc π 6 ( p6 ) = y1 = csc = 23 1 1 2 =2 ( p6 ) = 2 Got It? 1. What is the exact value of each expression? Do not use a calculator. p a. csc 3 ( ) 5p b. cot - 4 c. sec 3p d. Reasoning Use the unit circle at the right to find cot n, csc n, and sec n. Explain how you found your answers. 152 152 y n x 3 5 Chapter 13 Periodic Functions and Trigonometry HSM15_A2Hon_SE_CC_31_TrKit.indd 152 Common Core HSM15_A2 8 Use the reciprocal relationships to evaluate secant, cosecant, or cotangent on a calculator, since most calculators do not have these functions as menu options. Problem 2 Finding Values with a Calculator What is the decimal value of each expression? Use the radian mode on your calculator. Round to the nearest thousandth. A sec 2 Can you use the sin −1 , cos −1 , and tan −1 keys on the calculator for the reciprocal functions? No; those keys are inverse functions, not reciprocal functions. 2Hon_SE_CC_31_TrKit.indd 153 8/5/13 7:20 PM B cot 10 1 1 sec 2 = cos 2 1/cos(2) cot 10 = tan 10 1/tan(10) –2.402997962 sec 2 ≈ -2.403 1.542351045 cot 10 ≈ 1.542 C csc 35° D cot P csc 35° = 1 sin 35° cot p = tan1 p To evaluate an angle in degrees in radian mode, use the degree symbol from the ANGLE menu. 1/sin(35˚) ERR:DIVIDE BY 0 1:Quit 2:Goto 1.743446796 Evaluating cot p results in an error message, because tan p is equal to zero. csc 35° ≈ 1.743 Got It? 2. What is the decimal value of each expression? Use the radian mode on your calculator. Round your answers to the nearest thousandth. a. cot 13 b. csc 6.5 c. sec 15° 3p d. sec 2 e. Reasoning How can you find the cotangent of an angle without using the tangent key on your calculator? Lesson 13-8 Reciprocal Trigonometric Functions CC-31 Reciprocal Trigonometric Functions 153 8/5/13 153 7:20 PM The graphs of reciprocal trigonometric functions have asymptotes where the functions are undefined. Problem 3 Sketching a Graph For what values is csc x undefined? Wherever sin x = 0, its reciprocal is undefined. What are the graphs of y = sin x and y = csc x in the interval from 0 to 2P? Step 1 Make a table of values. x 0 sin x 0 0.5 csc x — 2 Step 2 p 3 p 2 2p 3 0.9 1.2 1 1 0.9 0.5 0 0.5 0.9 1 0.9 0.5 0 1.2 2 — 2 1.2 1 1.2 2 — p 6 5p 6 p 7p 6 4p 3 3p 2 5p 3 11p 2p 6 Plot the points and sketch the graphs. 2 y csc x 1 O 1 y = csc x will have a vertical asymptote whenever its denominator (sin x) is 0. y y sin x p x 2 Got It? 3. What are the graphs of y = tan x and y = cot x in the interval from 0 to 2p? You can use a graphing calculator to graph trigonometric functions quickly. Problem 4 Using Technology to Graph a Reciprocal Function How can you find the value? Use the table feature of your calculator. Graph y = sec x. What is the value of sec 20°? Step 1 Step 2 Use degree mode. 1 Graph y = cos x. Xmin –360 Xmax 360 Xscl 30 Ymin –5 Ymax 5 Yscl 1 Use the TABLE feature. sec 20° ≈ 1.0642 X 20 21 22 23 24 25 26 X = 20 Y1 1.0642 1.0711 1.0785 1.0864 1.0946 1.1034 1.1126 Got It? 4. What is the value of csc 45°? Use the graph of the reciprocal trigonometric function. 154 154 Chapter 13 Periodic Functions and Trigonometry HSM15_A2Hon_SE_CC_31_TrKit.indd 154 Common Core HSM15_A2 8 You can use a reciprocal trigonometric function to solve a real-world problem. Problem 5 Using Reciprocal Functions to Solve a Problem A restaurant is near the top of a tower. A diner looks down at an object along a line of sight that makes an angle of U with the tower. The distance in feet of an object from the observer is modeled by the function d = 601 sec U. How far away are objects sighted at angles d of 40° and 70°? U 601 ft Set your calculator to degree mode. Enter the function and construct a table that gives values of d for various angles of u. How can you check that your answers are correct? Multiply the answers by cos u. If the answers are correct, then the product is 601. Plot1 Plot2 \Y1 = 601/cos(X) \Y2 = \Y3 = \Y4 = \Y5 = \Y6 = \Y7 = Plot3 TABLE SETUP TblStart = 20 Tbl = 10 Indpnt: Auto Ask Depend: Auto Ask X 20 30 40 50 60 70 80 Y1 639.57 693.98 784.55 934.99 1202 1757.2 3461 X = 20 From the table, the objects are about 785 feet away and 1757 feet away, respectively. Got It? 5. The 601 in the function for Problem 5 is the diner’s height above the ground in feet. If the diner is 553 feet above the ground, how far away are objects sighted at angles of 50° and 80°? Lesson Check Do you know HOW? Do you UNDERSTAND? Find each value without using a calculator. p 1. csc 2 p 2. sec 1- 6 2 Use a calculator to find each value. Round your answers to the nearest thousandth. 3. csc 1.5 4. sec 42° 5. An extension ladder leans against a building forming a 50° angle with the ground. Use the function y = 21 csc x + 2 to find y, the length of the ladder. Round to the nearest tenth of a foot. 2Hon_SE_CC_31_TrKit.indd 155 8/5/13 7:20 PM MATHEMATICAL PRACTICES 6. Reasoning Explain why the graph of y = 5 sec u has no zeros. 7. Error Analysis On a quiz, a student wrote sec 20° + 1 = 0.5155. The teacher marked it wrong. What error did the student make? 8. Compare and Contrast How are the graphs of y = sec x and y = csc x alike? How are they different? Could the graph of y = csc x be a transformation of the graph of y = sec x? Lesson 13-8 Reciprocal Trigonometric Functions CC-31 Reciprocal Trigonometric Functions 155 8/5/13 155 7:20 PM Practice and Problem-Solving Exercises A Practice MATHEMATICAL PRACTICES See Problem 1. Find each value without using a calculator. If the expression is undefined, write undefined. 9. sec ( -p) ( 3p ) 13. cot - 2 ( p) ( ) 5p p 11. cot - 3 3p 15. sec - 4 10. csc 4 7p 14. csc 6 12. sec 2 16. cot ( -p) See Problem 2. Graphing Calculator Use a calculator to find each value. Round your answers to the nearest thousandth. 17. sec 2.5 18. csc ( -0.2) 19. cot 56° 21. cot ( -32°) 22. sec 195° 23. csc 0 ( 3p ) 20. sec - 2 24. cot ( -0.6) See Problem 3. Graph each function in the interval from 0 to 2P. 25. y = sec 2u 26. y = cot u 27. y = csc 2u - 1 28. y = csc 2u Graphing Calculator Use the graph of the appropriate reciprocal trigonometric function to find each value. Round to four decimal places. 29. sec 30° 30. sec 80° 31. sec 110° 32. csc 30° 33. csc 70° 34. csc 130° 35. cot 30° 36. cot 60° 37. Distance A woman looks out a window of a building. She is 94 feet above the ground. Her line of sight makes an angle of u with the building. The distance in feet of an object from the woman is modeled by the function d = 94 sec u. How far away are objects sighted at angles of 25° and 55°? B Apply See Problem 4. See Problem 5. 38. Think About a Plan A communications tower has wires anchoring it to the ground. Each wire is attached to the tower at a height 20 ft above the ground. The length y of the wire is modeled with the function y = 20 csc u, where u is the measure of the angle formed by the wire and the ground. Find the length of wire needed to form an angle of 45°. • Do you need to graph the function? • How can you rewrite the function so you can use a calculator? 39. Multiple Representations Write a cosecant model that has the same graph as y = sec u. Match each function with its graph. 40. y = sin1 x 1 41. y = cos x a. 156 156 b. 42. y = - sin1 x c. Chapter 13 Periodic Functions and Trigonometry HSM15_A2Hon_SE_CC_31_TrKit.indd Common156Core HSM15_A2 8/5/ 2Hon_SE_CC_31_TrKit.indd 157 /13 7:20 PM Graph each function in the interval from 0 to 2P. p 1 u 45. y = -sec pu 43. y = csc u - 2 44. y = sec 4 u 47. a. What are the domain, range, and period of y = csc x? b. What is the relative minimum in the interval 0 … x … p? c. What is the relative maximum in the interval p … x … 2p? 46. y = cot 3 48. Reasoning Use the relationship csc x = sin1 x to explain why each statement is true. a. When the graph of y = sin x is above the x-axis, so is the graph of y = csc x. b. When the graph of y = sin x is near a y-value of -1, so is the graph of y = csc x. Writing Explain why each expression is undefined. 49. csc 180° 50. sec 90° 51. cot 0° 52. Indirect Measurement The fire ladder forms an angle of measure u with the horizontal. The hinge of the ladder y is 35 ft from the building. The function y = 35 sec u models the length y in feet that the fire ladder must be 35 ft to reach the building. 8 ft a. Graph the function. b. In the photo, u = 13°. What is the ladder’s length? c. How far is the ladder extended when it forms an angle of 30°? d. Suppose the ladder is extended to its full length of 80 ft. What angle does it form with the horizontal? How far up a building can the ladder reach when fully extended? (Hint: Use the information in the photo.) θ 53. a. Graph y = tan x and y = cot x on the same axes. b. State the domain, range, and asymptotes of each function. c. Compare and Contrast Compare the two graphs. How are they alike? How are they different? d. Geometry The graph of the tangent function is a reflection image of the graph of the cotangent function. Name at least two reflection lines for such a transformation. Graphing Calculator Graph each function in the interval from 0 to 2P. Describe any phase shift and vertical shift in the graph. 54. y = sec 2u + 3 57. f (x) = 3 csc (x + 2) - 1 55. y = sec 2 1 u + 2 2 58. y = cot 2(x + p) + 3 p 56. y = -2 sec (x - 4) 59. g (x) = 2 sec 1 3 1 x - 6 22 - 2 p Graph y = -cos x and y = -sec x on the same axes. State the domain, range, and period of each function. For which values of x does -cos x = -sec x? Justify your answer. Compare and Contrast Compare the two graphs. How are they alike? How are they different? e. Reasoning Is the value of -sec x positive when -cos x is positive and negative when -cos x is negative? Justify your answer. 60. a. b. c. d. Lesson 13-8 Reciprocal Trigonometric Functions CC-31 Reciprocal Trigonometric Functions 157 8/5/13 157 7:20 PM 61. a. Reasoning Which expression gives the correct value of csc 60°? I. sin ((60-1)°) C Challenge II. (sin 60°)-1 III. (cos 60°)-1 1 ° b. Which expression in part (a) represents sin 1 60 2? 62. Reasoning Each branch of y = sec x and y = csc x is a curve. Explain why these curves cannot be parabolas. (Hint: Do parabolas have asymptotes?) 63. Reasoning Consider the relationship between the graphs of y = cos x and y = cos 3x. Use the relationship to explain the distance between successive branches of the graphs of y = sec x and y = sec 3x. 64. a. Graph y = cot x, y = cot 2x, y = cot ( -2x), and y = cot 12x on the same axes. b. Make a Conjecture Describe how the graph of y = cot bx changes as the value of b changes. 158 158 Chapter 13 Periodic Functions and Trigonometry HSM15_A2Hon_SE_CC_31_TrKit.indd Common158Core 8/5/