## Problem Set 47: Right Triangle Trigonometry

1. For the given right triangle, label the adjacent side, opposite side, and hypotenuse for the indicated angle.

2. When a right triangle with a hypotenuse of 1 is placed in the unit circle, which sides of the triangle correspond to the x– and y-coordinates?

3. The tangent of an angle compares which sides of the right triangle?

4. What is the relationship between the two acute angles in a right triangle?

5. Explain the cofunction identity.

For the following exercises, use cofunctions of complementary angles.

6. $\cos \left(34^\circ\right)=\sin \left(\text{__}^\circ\right)$

7. $\cos \left(\frac{\pi }{3}\right)=\sin \text{(___)}$

8. $\csc \left(21^\circ\right)=\sec \left(\text{___}^\circ \right)$

9. $\tan \left(\frac{\pi }{4}\right)=\cot \left(\text{__}\right)$

For the following exercises, find the lengths of the missing sides if side $a$ is opposite angle $A$, side $b$ is opposite angle $B$, and side $c$ is the hypotenuse.

10. $\cos B=\frac{4}{5},a=10$

11. $\sin B=\frac{1}{2}, a=20$

12. $\tan A=\frac{5}{12},b=6$

13. $\tan A=100,b=100$

14. $\sin B=\frac{1}{\sqrt{3}}, a=2$

15. $a=5,\measuredangle A={60}^{\circ }$

16. $c=12,\measuredangle A={45}^{\circ }$

For the following exercises, use Figure 14 to evaluate each trigonometric function of angle $A$.

17. $\sin A$

18. $\cos A$

19. $\tan A$

20. $\csc A$

21. $\sec A$

22. $\cot A$

For the following exercises, use Figure 15 to evaluate each trigonometric function of angle $A$.

23. $\sin A$

24. $\cos A$

25. $\tan A$

26. $\csc A$

27. $\sec A$

28. $\cot A$

For the following exercises, solve for the unknown sides of the given triangle.

29.

30.

31.

For the following exercises, use a calculator to find the length of each side to four decimal places.

32.

33.

34.

35.

36.

37. $b=15,\measuredangle B={15}^{\circ }$

38. $c=200,\measuredangle B={5}^{\circ }$

39. $c=50,\measuredangle B={21}^{\circ }$

40. $a=30,\measuredangle A={27}^{\circ }$

41. $b=3.5,\measuredangle A={78}^{\circ }$

42. Find $x$.

43. Find $x$.

44. Find $x$.

45. Find $x$.

46. A radio tower is located 400 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is $36^\circ$, and that the angle of depression to the bottom of the tower is $23^\circ$. How tall is the tower?

47. A radio tower is located 325 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is $43^\circ$, and that the angle of depression to the bottom of the tower is $31^\circ$. How tall is the tower?

48. A 200-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is $15^\circ$, and that the angle of depression to the bottom of the tower is $2^\circ$. How far is the person from the monument?

49. A 400-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is $18^\circ$, and that the angle of depression to the bottom of the tower is $3^\circ$. How far is the person from the monument?

50. There is an antenna on the top of a building. From a location 300 feet from the base of the building, the angle of elevation to the top of the building is measured to be $40^\circ$. From the same location, the angle of elevation to the top of the antenna is measured to be $43^\circ$. Find the height of the antenna.

51. There is lightning rod on the top of a building. From a location 500 feet from the base of the building, the angle of elevation to the top of the building is measured to be $36^\circ$. From the same location, the angle of elevation to the top of the lightning rod is measured to be $38^\circ$. Find the height of the lightning rod.

52. A 33-ft ladder leans against a building so that the angle between the ground and the ladder is $80^\circ$. How high does the ladder reach up the side of the building?

53. A 23-ft ladder leans against a building so that the angle between the ground and the ladder is $80^\circ$. How high does the ladder reach up the side of the building?

54. The angle of elevation to the top of a building in New York is found to be 9 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.

55. The angle of elevation to the top of a building in Seattle is found to be 2 degrees from the ground at a distance of 2 miles from the base of the building. Using this information, find the height of the building.

56. Assuming that a 370-foot tall giant redwood grows vertically, if I walk a certain distance from the tree and measure the angle of elevation to the top of the tree to be $60^\circ$, how far from the base of the tree am I?