CWA CH 13 FINAL_2_97_22 - Chapter 13 THE TRIGONOMETRIC...

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Chapter 13 THE TRIGONOMETRIC FUNCTIONS 13.1 De…nitions of the Trigonometric Functions 1. 60 ± =60 ³ ¼ 180 ´ = ¼ 3 2. 90 ± =90 ³ ¼ 180 ´ = ¼ 2 3. 150 ± =150 ³ ¼ 180 ´ = 5 ¼ 6 4. 135 ± =135 ³ ¼ 180 ´ = 3 ¼ 4 5. 270 ± =270 ³ ¼ 180 ´ = 3 ¼ 2 6. 320 ± =320 ³ ¼ 180 ´ = 16 ¼ 9 7. 495 ± =495 ³ ¼ 180 ´ = 11 ¼ 4 8. 510 ± =510 ³ ¼ 180 ´ = 17 ¼ 6 9. 5 ¼ 4 = 5 ¼ 4 μ 180 ± ¼ =225 ± 10. 2 ¼ 3 = 2 ¼ 3 μ 180 ± ¼ =120 ± 11. ¡ 13 ¼ 6 = ¡ 13 ¼ 6 μ 180 ± ¼ = ¡ 390 ± 12. ¡ ¼ 4 = ¡ ¼ 4 μ 180 ± ¼ = ¡ 45 ± 13. 8 ¼ 5 = 8 ¼ 5 μ 180 ± ¼ =288 ± 14. 5 ¼ 9 = 5 ¼ 9 μ 180 ± ¼ =100 ± 15. 7 ¼ 12 = 7 ¼ 12 μ 180 ± ¼ =105 ± 16. 5 ¼ =5 ¼ μ 180 ± ¼ =900 ± 17. Let ® = the angle with terminal side through ( ¡ 3 ; 4) : Then x = ¡ 3 ;y =4 ; and r = p x 2 + y 2 = p ( ¡ 3) 2 +(4) 2 = p 25 = 5 : sin ® = y r = 4 5 cot ® = x y = ¡ 3 4 cos ® = x r = ¡ 3 5 sec ® = r x = ¡ 5 3 tan ® = y x = ¡ 4 3 csc ® = r y = 5 4 18. Let ® = the angle with terminal side through ( ¡ 12 ; ¡ 5) : Then x = ¡ 12 = ¡ 5 ; and r = p x 2 + y 2 = p 144 + 25 = p 169 = 13 : sin ® = y r = ¡ 5 13 cot ® = x y = 12 5 cos ® = x r = ¡ 12 13 sec ® = r x = ¡ 13 12 tan ® = y x = 5 12 csc ® = r y = ¡ 13 5 19. Let ® = the angle with terminal side through (7 ; ¡ 24) : Then x =7 = ¡ 24 ; and r = p x 2 + y 2 = p 49 + 576 = p 625 = 25 : sin ® = y r = ¡ 24 25 cot ® = x y = ¡ 7 24 cos ® = x r = 7 25 sec ® = r x = 25 7 tan ® = y x = ¡ 24 7 csc ® = r y = ¡ 25 24 20. Let ® = the angle with terminal side through (20 ; 15) : Then x =20 =15 ; and r = p x 2 + y 2 = p 400 + 225 = p 625 = 25 : sin ® = y r = 3 5 cot ® = x y = 4 3 cos ® = x r = 4 5 sec ® = r x = 5 4 tan ® = y x = 3 4 csc ® = r y = 5 3 853
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854 Chapter 13 THE TRIGONOMETRIC FUNCTIONS 21. In quadrant I, all six trigonometric functions are positive, so their sign is +. 22. In quadrant II, x< 0 and y> 0 : Furthermore, r> 0 : sin μ = y r > 0 ; so the sign is +. cos μ = x r < 0 ; so the sign is ¡ : tan μ = y x < 0 ; so the sign is ¡ : cot μ = x y < 0 ; so the sign is ¡ : sec μ = r x < 0 ; so the sign is ¡ : csc μ = r y > 0 ; so the sing is +. 23. In quadrant III, 0 and y< 0 : Furthermore, 0 : sin μ = y r < 0 ; so the sign is ¡ : cos μ = x r < 0 ; so the sign is ¡ : tan μ = y x > 0 ; so the sign is +. cot μ = x y > 0 ; so the sign is +. sec μ = r x < 0 ; so the sign is ¡ : csc μ = r y < 0 ; so the sign is ¡ : 24. In quadrant IV, x> 0 and 0 : Also, 0 : sin μ = y r < 0 ; so the sign is ¡ . cos μ = x r > 0 ; so the sign is + : tan μ = y x < 0 ; so the sign is ¡ : cot μ = x y < 0 ; so the sign is ¡ : sec μ = r x > 0 ; so the sign is + : csc μ = r y < 0 ; so the sign is ¡ . 25. When an angle μ of 30 ± is drawn in standard po- sition, one choice of a point on its terminal side is ( x;y )=( p 3 ; 1) : Then r = p x 2 + y 2 = p 3+1=2 : tan μ = y x = 1 p 3 = p 3 3 cot μ = x y = p 3 csc μ = r y =2 26. When an angle μ of 45 ± is drawn in standard po- sition, ( )=(1 ; 1) is one point on its terminal side. Then r = p 1+1= p 2 : sin μ = y r = 1 p 2 = p 2 2 cos μ = x r = 1 p 2 = p 2 2 sec μ = r x = p 2 csc μ = r y = p 2 27. When an angle μ of 60 ± is drawn in standard po- sition, one choice of a point on its terminal side is ( ; p 3) : Then r = p x 2 + y 2 = p 1+3=2 : sin μ = y r = p 3 2 cot μ = x y = 1 p 3 = p 3 3 csc μ = r y = 2 p 3 = 2 p 3 3 28. When an angle μ of 120 ± is drawn in standard posi- tion, ( ¡ 1 ; p 3) is one point on its terminal side. Then r = p : cos μ = x r = ¡ 1 2 cot μ = x y = ¡ 1 p 3 = ¡ p 3 3 sec μ = r x = ¡ 2
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Section 13.1 De…nitions of the Trigonometric Functions 855 29. When an angle μ of 135 ± is drawn in standard
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This note was uploaded on 10/07/2010 for the course CALC 2302 taught by Professor Yuly during the Fall '08 term at University of Texas at Dallas, Richardson.

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CWA CH 13 FINAL_2_97_22 - Chapter 13 THE TRIGONOMETRIC...

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