Problem Set 47: Right Triangle Trigonometry

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

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. [latex]\cos \left(34^\circ\right)=\sin \left(\text{__}^\circ\right)[/latex]

7. [latex]\cos \left(\frac{\pi }{3}\right)=\sin \text{(___)}[/latex]

8. [latex]\csc \left(21^\circ\right)=\sec \left(\text{___}^\circ \right)[/latex]

9. [latex]\tan \left(\frac{\pi }{4}\right)=\cot \left(\text{__}\right)[/latex]

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

10. [latex]\cos B=\frac{4}{5},a=10[/latex]

11. [latex]\sin B=\frac{1}{2}, a=20[/latex]

12. [latex]\tan A=\frac{5}{12},b=6[/latex]

13. [latex]\tan A=100,b=100[/latex]

14. [latex]\sin B=\frac{1}{\sqrt{3}}, a=2[/latex]

15. [latex]a=5,\measuredangle A={60}^{\circ }[/latex]

16. [latex]c=12,\measuredangle A={45}^{\circ }[/latex]

For the following exercises, use Figure 14 to evaluate each trigonometric function of angle [latex]A[/latex].

A right triangle with sides 4 and 10 and angle of A labeled.

Figure 14

17. [latex]\sin A[/latex]

18. [latex]\cos A[/latex]

19. [latex]\tan A[/latex]

20. [latex]\csc A[/latex]

21. [latex]\sec A[/latex]

22. [latex]\cot A[/latex]

For the following exercises, use Figure 15 to evaluate each trigonometric function of angle [latex]A[/latex].

A right triangle with sides of 10 and 8 and angle of A labeled.

Figure 15

23. [latex]\sin A[/latex]

24. [latex]\cos A[/latex]

25. [latex]\tan A[/latex]

26. [latex]\csc A[/latex]

27. [latex]\sec A[/latex]

28. [latex]\cot A[/latex]

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

29.
A right triangle with sides of 7, b, and c labeled. Angles of B and 30 degrees also labeled.

30.
A right triangle with sides of 10, a, and c. Angles of 60 degrees and A also labeled.

31.
A right triangle with corners labeled A, B, and C. Hypotenuse has length of 15 times square root of 2. Angle B is 45 degrees.

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

32.
A right triangle with sides of 10, a, and c. Angles of A and 62 degrees are also labeled.

33.
A right triangle with sides of 7, b, and c. Angles of 35 degrees and B are also labeled.

34.
A right triangle with sides of a, b, and 10 labeled. Angles of 65 degrees and B are also labeled.

35.
A right triangle with sides a, b, and 12. Angles of 10 degrees and B are also labeled.

36.
A right triangle with corners labeled A, B, and C. Sides labeled b, c, and 16.5. Angle of 81 degrees also labeled.

37. [latex]b=15,\measuredangle B={15}^{\circ }[/latex]

38. [latex]c=200,\measuredangle B={5}^{\circ }[/latex]

39. [latex]c=50,\measuredangle B={21}^{\circ }[/latex]

40. [latex]a=30,\measuredangle A={27}^{\circ }[/latex]

41. [latex]b=3.5,\measuredangle A={78}^{\circ }[/latex]

42. Find [latex]x[/latex].
A triangle with angles of 63 degrees and 39 degrees and side x. Bisector in triangle with length of 82.

43. Find [latex]x[/latex].
A triangle with angles of 36 degrees and 50 degrees and side x. Bisector in triangle with length of 85.

44. Find [latex]x[/latex].
A right triangle with side of 115 and angle of 35 degrees. Within right triangle there is another right triangle with angle of 56 degrees. Side length difference between two triangles is x.

45. Find [latex]x[/latex].
A right triangle with side of 119 and angle of 26 degrees. Within right triangle there is another right triangle with angle of 70 degrees instead of 26 degrees. Difference in side length between two triangles is 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 [latex]36^\circ [/latex], and that the angle of depression to the bottom of the tower is [latex]23^\circ [/latex]. 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 [latex]43^\circ [/latex], and that the angle of depression to the bottom of the tower is [latex]31^\circ [/latex]. 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 [latex]15^\circ [/latex], and that the angle of depression to the bottom of the tower is [latex]2^\circ [/latex]. 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 [latex]18^\circ [/latex], and that the angle of depression to the bottom of the tower is [latex]3^\circ [/latex]. 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 [latex]40^\circ [/latex]. From the same location, the angle of elevation to the top of the antenna is measured to be [latex]43^\circ [/latex]. 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 [latex]36^\circ [/latex]. From the same location, the angle of elevation to the top of the lightning rod is measured to be [latex]38^\circ [/latex]. 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 [latex]80^\circ [/latex]. 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 [latex]80^\circ [/latex]. 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 [latex]60^\circ [/latex], how far from the base of the tree am I?