3.1 Section Exercises

For Exercises 1-10, find the values of all six trigonometric functions of angles A and B in the right triangle △ ABC in Figure 8.

Figure 8. A right triangle with angles A B C and sides a b c

1. [latex] a = 5 [/latex], [latex] b = 12 [/latex], [latex] c = 13 [/latex]
2. [latex] a = 8 [/latex], [latex] b = 15 [/latex], [latex] c = 17 [/latex]
3. [latex] a = 7 [/latex], [latex] b = 24 [/latex], [latex] c = 25 [/latex]
4. [latex] a = 20 [/latex], [latex] b = 21 [/latex], [latex] c = 29 [/latex]
5. [latex] a = 9 [/latex], [latex] b = 40 [/latex], [latex] c = 41 [/latex]
6. [latex] a = 1 [/latex], [latex] b = 2 [/latex], [latex] c = \sqrt{5} [/latex]
7. [latex] a = 1 [/latex], [latex] b = 3 [/latex]
8. [latex] a = 2 [/latex], [latex] b = 5 [/latex]
9. [latex] a = 5 [/latex], [latex] c = 6 [/latex]
10. [latex] b = 7 [/latex], [latex] c = 8 [/latex]

For Exercises 11-18, find the values of the other five trigonometric functions of the acute angle A given the indicated value of one of the functions.

11. [latex] \sin\;A = \frac{3}{4} [/latex]
12. [latex] \cos\;A = \frac{2}{3} [/latex]
13. [latex] \cos\;A = \frac{2}{\sqrt{10}} [/latex]
14. [latex] \sin\;A = \frac{2}{4} [/latex]
15. [latex] \tan\;A = \frac{5}{9} [/latex]
16. [latex] \tan\;A = 3 [/latex]
17. [latex] \sec\;A = \frac{7}{3} [/latex]
18. [latex] \csc\;A = 3 [/latex]

For Exercises 19-23, write the given number as a trigonometric function of an acute angle less than 45° .

19. [latex] \sin\;87^\circ [/latex]
20. [latex] \sin\;53^\circ [/latex]
21. [latex] \cos\;46^\circ [/latex]
22. [latex] \tan\;66^\circ [/latex]
23. [latex] \sec\;77^\circ [/latex]

For Exercises 24-28, write the given number as a trigonometric function of an acute angle greater than 45[latex] ^\circ [/latex] .

24. [latex] \sin\;1^\circ [/latex]
25. [latex] \cos\;13^\circ [/latex]
26. [latex] \tan\;26^\circ [/latex]
27. [latex] \cot\;10^\circ [/latex]
28. [latex] \csc\;43^\circ [/latex]
29. In Example 1.7 we found the values of all six trigonometric functions of 60° and 30° .
    1. Does [latex] \;\sin\;30^\circ ~+~ \sin\;30^\circ ~=~ \sin\;60^\circ [/latex]?
    2. Does [latex] \;\cos\;30^\circ ~+~ \cos\;30^\circ ~=~ \cos\;60^\circ [/latex]?
    3. Does [latex] \;\tan\;30^\circ ~+~ \tan\;30^\circ ~=~ \tan\;60^\circ [/latex]?
    4. Does [latex] \;2\,\sin\;30^\circ\,\cos\;30^\circ ~=~ \sin\;60^\circ [/latex]?
30. For an acute angle [latex] A [/latex], can [latex] \sin\;A [/latex] be larger than [latex] 1 [/latex]? Explain your answer.
31. For an acute angle [latex] A [/latex], can [latex] \cos\;A [/latex] be larger than [latex] 1 [/latex]? Explain your answer.
32. For an acute angle [latex] A [/latex], can [latex] \sin\;A [/latex] be larger than [latex] \tan\;A [/latex]? Explain your answer.
33. If [latex] A [/latex] and [latex] B [/latex] are acute angles and [latex] A < B [/latex], explain why [latex] \sin\;A < \sin\;B [/latex].
34. If [latex] A [/latex] and [latex] B [/latex] are acute angles and [latex] A < B [/latex], explain why [latex] \cos\;A > \cos\;B [/latex].
35. Prove the Cofunction Theorem. Hint: Draw a right triangle and label the angles and sides.