Exercise Set 4.4: Trigonometric Expressions and Identities

Exercise Set 4.4: Trigonometric Expressions and Identities

Answer the following.

1. Beginning with the Pythagorean identity

cos2 ( ) + sin2 ( ) = 1 , establish another

Pythagorean identity by dividing each term by

cos2 ( ) . Show all work.

2. Beginning with the Pythagorean identity

cos2 ( ) + sin2 ( ) = 1 , establish another

Pythagorean identity by dividing each term by

sin2 ( ) . Show all work.

Solve the following algebraically, using identities from this section.

3. If cos ( ) = 5 and 3 < < 2 , find the exact

13

2

values of sin ( ) and tan ( ) .

4. If sin ( ) = 4 and < < , find the exact

52

values of cos ( ) and tan ( ) .

5. If sin ( ) = - 1 and 180D < < 270D , find the

9 five remaining trigonometric functions of .

6. If cos ( ) = 1 and 270D < < 360D , find the

10 five remaining trigonometric functions of .

7. If csc ( ) = - 9 and 3 < < 2 , find the exact

4

2

values of cot ( ) and sin ( ) .

8. If sec( ) = - 5 and < < 3 , find the exact

2

2

values of tan ( ) and cos ( ) .

9. If cot ( ) = - 13 and 90D < < 180D , find the

6

exact values of csc( ) and sin ( ) .

10. If tan ( ) = 1 and 180D < < 270D , find the

3

exact values of sec( ) and cot ( ) .

11. If tan ( ) = 3 and < < 3 , find the five

4

2

remaining trigonometric functions of .

Math 1330, Precalculus The University of Houston

12. If cot ( ) = - 12 and < < , find the five

5

2

remaining trigonometric functions of .

Perform the following operations and combine like terms. (Do not rewrite the terms or the solution in terms of any other trigonometric functions.)

13. sin ( ) - 3 sin ( ) + 5

14. tan ( ) - 7 tan ( ) - 2

15. csc( x) - 32

16. 3 + sec( x)2

17.

3

cos (

)

-

7

sin (

)

18.

-

6

sin (

)

+

2

cos (

)

19. 5 tan ( )sin ( ) - 8 tan ( )sin ( )

20. 8cos ( ) cot ( ) + 9cos ( ) cot ( )

Factor each of the following expressions.

21. 25sin2 ( ) - 49 cos2 ( ) 22. 16 cos2 ( ) - 81sin2 ( ) 23. sec2 ( ) - 7 sec( ) +12 24. tan2 ( ) + 9 tan ( ) +12 25. 10 cot2 ( ) -13cot ( ) - 3 26. 6 csc2 ( ) + 7 csc( ) - 5

Simplify the following.

27. cos ( ) tan ( ) 28. sin ( )cot ( ) 29. csc2 ( x) tan2 ( x) 30. sec3 ( x) cot3 ( x)

Chapter 4: Trigonometric Functions

Exercise Set 4.4: Trigonometric Expressions and Identities

31. cot ( x) csc ( x) tan2 ( x)

32. sin ( x)sec( x) cot2 ( x)

33. 1- sin2 ( x)

34. cos2 ( x) -1

35. sec ( ) -1 sec( ) +1 36. csc( x) +1 csc( x) -1 37. sin ( ) csc( ) - sin ( )

38. cos ( ) sec( ) - cos ( )

39. sec2 ( ) -1 csc2 ( ) -1

1- sin2 ( x) 40. cos2 ( x) -1

41. sin ( x) cot ( x) + tan ( x) 42. cos ( x) tan ( x) + cot ( x)

43. sec( x) - tan ( x) sec( x) + tan ( x)

44. csc( x) - cot ( x) csc( x) + cot ( x)

45. 1- cos ( x) csc( x) + cot ( x)

46. csc( x) -1 sec( x) + tan ( x)

cot ( x) + tan ( x)

47.

csc2 ( x)

sec2 ( x) 48. tan ( x) + cot ( x)

cos ( x) cos ( x) 49. 1- sin ( x) - 1+ sin ( x)

cot ( ) cot ( ) 50. csc( ) +1 + csc( ) -1

cos2 ( ) - sin2 ( ) 51. sin4 ( ) - cos4 ( )

Math 1330, Precalculus The University of Houston

sec4 ( x) - tan4 ( x) 52. sec2 ( x) + tan2 ( x)

Prove each of the following identities.

53. sec( x) - sin ( x) tan ( x) = cos ( x)

54. cos ( x) cot ( x) - csc( x) = - sin ( x)

55.

sec( )csc( ) tan ( ) + cot ( )

=

1

56. sin ( ) - cos ( )2 + sin ( ) + cos ( )2 = 2 57. tan ( x) - sin ( x) cos ( x) = tan ( x)sin2 ( x)

58. cot ( x) - cos ( x)sin ( x) = cot ( x) cos2 ( x)

59. cot2 ( x) - cos2 ( x) = cot2 ( x) cos2 ( x)

60. tan2 ( x) - sin2 ( x) = tan2 ( x)sin2 ( x)

61.

cot ( x) - tan ( x) sin ( x)cos ( x)

=

csc2

(

x

)

-

sec2

(

x)

62.

1

tan ( x + sec(

)

x

)

+

1

+ sec( x) tan ( x)

=

2

csc

(

x)

63. 2 cos2 ( x) + 5sin2 ( x) = 2 + 3sin2 ( x) 64. 3sin2 ( x) - 4 cos2 ( x) = 3 - 7 cos2 ( x) 65. tan4 ( ) + tan2 ( ) = sec4 ( ) - sec2 ( ) 66. csc4 ( x) - csc2 ( x) = cot4 ( x) + cot2 ( x)

67.

1

sin ( x) + cos ( x)

=

csc

(

x

)

-

cot

(

x

)

68.

1

cos ( x + sin (

) x)

=

sec

(

x

)

-

tan

(

x)

69.

1

cos ( x + sin (

) x)

=

1- sin ( x cos( x)

)

1- cos ( ) sin ( ) 70. sin ( ) = 1+ cos ( )

Chapter 4: Trigonometric Functions

Exercise Set 4.4: Trigonometric Expressions and Identities

Another method of solving problems like exercises 312 is shown below.

Let the terminal side of an angle intersect the circle x2 + y2 = r2 .

First draw the reference angle, , for . Then, draw a right triangle

with its hypotenuse extending from the origin to ( x, y) and its leg is on

the x-axis, as shown below. Label the legs x and y (which may be negative), and the hypotenuse with positive length r.

y

( x, y)

r

y

x

x

Answer the following.

77. If the terminal ray of an angle in standard

position passes through the point (12, - 5) , find

the six trigonometric functions of .

78. If the terminal ray of an angle in standard

position passes through the point (-8, 15) , find

the six trigonometric functions of .

79. If the terminal ray of an angle in standard

position passes through the point (-6, - 2) , find

the six trigonometric functions of .

80. If the terminal ray of an angle in standard

position passes through the point (3, - 9) , find

the six trigonometric functions of .

Then find the six trigonometric functions of , using right triangle trigonometry ( sin = the leg opposite angle , etc...) The six

the hypotenuse trigonometric functions of are the same as the six trigonometric

functions of . (This can be compared with the ratios sin = y , etc. at r

the beginning of the text in Section 4.3).

In each of the following examples, use the given information to sketch and label a right triangle in the appropriate quadrant, as described above. Use the Pythagorean Theorem to find the missing side, and then use right triangle trigonometry to evaluate the five remaining trigonometric functions of .

71. sin ( ) = - 1 and 270D < < 360D

4

72. cos ( ) = - 13 and 180D < < 270D

7

73. sec( ) = - 7 and < <

2

2

74. csc( ) = 5 and < <

2

75. tan ( ) = 2 and < < 3

5

2

76. cot ( ) = - 11 and 3 < < 2

5

2

Math 1330, Precalculus The University of Houston

Chapter 4: Trigonometric Functions

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