
# 5.2e: Exercises - Right Angle Trigonometry


### A: Given two sides of an acute angle, find all six trigonometric ratios

Exercise $$\PageIndex{A}$$

$$\bigstar$$ Find the values of all six trigonometric functions of angles $$A$$ and $$B$$ in the right triangle $$\triangle\,ABC$$ illustrated below. Simplify!

 1. $$b = 12$$, $$c = 13$$ 2. $$b = 15$$, $$c = 17$$ 3.  $$a = 7$$, $$c = 25$$ 4.  $$a = 20$$, $$c = 29$$ 5.  $$a = 9$$, $$b = 40$$ 6.  $$a = 1$$, $$b = 2$$ 7. $$a = 1$$, $$b = 3$$ 8. $$a = 2$$, $$b = 5$$ 9. $$a = 5$$, $$c = 6$$ 10. $$a = 2$$, $$c = \sqrt{6}$$ 11. $$b = 7$$, $$c = 8$$  12. $$b = 3$$, $$c = \sqrt{15}$$

$$\bigstar$$ In each of the triangles below, find $$\sin \left(A\right), \; \cos \left(A\right),\; \tan \left(A\right),\; \sec \left(A\right),\; \csc \left(A\right),\; \cot \left(A\right)$$.

17.     18.

For all triangles #1-12, the trigonometric ratios can be determined using the following formulas: $$\sin A = \cos B = \frac{a}{c},$$  $$\cos A = \sin B = \frac{b}{c},$$ $$\tan A = \cot B = \frac{a}{b},$$  $$\csc A = \sec B = \frac{c}{a},$$  $$\sec A = \csc B = \frac{c}{b},$$ $$\cot A = \tan B = \frac{b}{a},$$  The missing side ineeded to calculate these ratios is given below:
1. $$a = 5$$     3. $$b = 24$$     5. $$c = 41$$     7. $$c = \sqrt{10}$$     9. $$b = \sqrt{11}$$      11. a = $$\sqrt{15}$$
17. $$\sin \left(A\right) = \frac{5\sqrt{41}}{41}, \; \cos \left(A\right) = \frac{4\sqrt{41}}{41},\; \tan \left(A\right) = \frac{5}{4},\; \sec \left(A\right) = \frac{\sqrt{41}}{4},\; \csc \left(A\right)= \frac{\sqrt{41}}{5},\; \cot \left(A\right) = \frac{4}{5}$$

### B: Given one trig ratio of an acute angle, find the other trigonometric ratios

Exercise $$\PageIndex{B}$$

$$\bigstar$$ Find the values of the other five trigonometric functions of the acute angle $$A$$ given the indicated value of one of the functions. Simplify!

 21. $$\sin A= \dfrac{1}{2}$$ 22. $$\sin A= \dfrac{5}{12}$$ 23.  $$\cos\;A = \dfrac{1}{5}$$ 24.  $$\cos\;A = \dfrac{2}{3}$$ 25.  $$\tan\;A = \dfrac{3}{4}$$ 26.  $$\tan\;A = \dfrac{9}{5}$$ 27.  $$\csc\;A = \dfrac{7}{3}$$ 28. $$\csc A=\dfrac{\sqrt{3}}{1}$$ 29.  $$\sec\;A = 3$$ 30.  $$\sec\;A = \dfrac{5\sqrt{6}}{3}$$ 31. $$\cot A=100$$ 32.  $$\cot\;A = 4$$

21. $$\sin \theta = \frac{1}{2}$$,   $$\cos \theta = \frac{\sqrt{3}}{2}$$,    $$\tan \theta = \frac{\sqrt{3}}{3}$$,    $$\csc \theta = 2$$,    $$\sec \theta = \frac{2\sqrt{3}}{3}$$,   $$\cot \theta = \sqrt{3}$$
23. $$\sin \theta = \frac{2\sqrt{6}}{5}$$,    $$\cos \theta = \frac{1}{5}$$,    $$\tan \theta = 2\sqrt{6}$$,    $$\csc \theta = \frac{5\sqrt{6}}{12}$$,    $$\sec \theta = 5$$,    $$\cot \theta = \frac{\sqrt{6}}{12}$$
25. $$\sin \theta = \frac{3}{5}$$,    $$\cos \theta = \frac{4}{5}$$,    $$\tan \theta = \frac{3}{4}$$,    $$\csc \theta = \frac{5}{3}$$,    $$\sec \theta = \frac{5}{4}$$,    $$\cot \theta = \frac{4}{3}$$
27. $$\sin \theta = \frac{3}{7}$$,    $$\cos \theta = \frac{2\sqrt{10}}{7}$$,    $$\tan \theta = \frac{3\sqrt{10}}{20}$$,    $$\csc \theta = \frac{7}{3}$$,    $$\sec \theta = \frac{7\sqrt{10}}{20}$$,    $$\cot \theta = \frac{2\sqrt{10}}{3}$$
29. $$\sin \theta = \frac{2\sqrt{2}}{3}$$,    $$\cos \theta = \frac{1}{3}$$,    $$\tan \theta = 2\sqrt{2}$$,    $$\csc \theta = \frac{3\sqrt{2}}{4}$$,    $$\sec \theta = 3$$,    $$\cot \theta = \frac{\sqrt{2}}{4}$$
31. $$\sin \theta = \frac{\sqrt{101}}{101}$$,    $$\cos \theta = \frac{100\sqrt{101}}{101}$$,    $$\tan \theta = \frac{1}{100}$$,    $$\csc \theta =\sqrt{101}$$,    $$\sec \theta = \frac{\sqrt{101}}{100}$$,    $$\cot \theta = 100$$

### D: Cofunctions

Exercise $$\PageIndex{D}$$

Use cofunctions of complementary angles to write an equivalent expression.

 36. $$\cos (34°)= \sin (\_\_°)$$ 37. $$\cos (\dfrac{π}{3})= \sin (\_\_\_)$$ 38. $$\csc (21°) = \sec (\_\_\_°)$$ 39. $$\tan (\dfrac{π}{4})= \cot (\_\_)$$

37. $$\dfrac{π}{6}$$       39. $$\dfrac{π}{4}$$

### C: Given one side and and an acute angle for a right triangle, find the other sides and angles

Exercise $$\PageIndex{A}$$

$$\bigstar$$ In each of the following triangles, solve for the unknown sides and angles. Give answers to 4 decimal digits.

41.                42.             43.

44.         45.          46.

47.                   48.

49. $$a=5,$$ angle opposite side $$a$$ is $$∡ A=60^∘$$

50. Hypotenuse $$c=12,$$ and one acute angle is $$∡ A=45^∘$$

$$\bigstar$$ Find $$x$$.

51.          52.

53.         54.

41. $$b \approx 12.1244$$, $$c =14$$, $$B = 60^{\circ}$$    43. $$a \approx 5.3171$$, $$c \approx 11.3257$$, $$A = 28^{\circ}$$    45. $$a \approx 9.0631$$, $$b \approx 4.2262$$, $$B = 25^{\circ}$$
47. $$a = 15$$, $$b = 15$$, $$A = 45^{\circ}$$    49. $$b \approx 2.8868$$, $$c \approx 5.7735$$, $$B = 30^{\circ}$$    51. $$188.3159$$     53. $$200.6737$$

### E: Applications

Exercise $$\PageIndex{E}$$

60. 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 $$36°$$, and that the angle of depression to the bottom of the tower is $$23°$$. How tall is the tower?

61. 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 $$43°$$, and that the angle of depression to the bottom of the tower is $$31°$$. How tall is the tower?

62. 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 $$15°$$, and that the angle of depression to the bottom of the tower is $$2°$$. How far is the person from the monument?

63. 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 $$18°$$, and that the angle of depression to the bottom of the monument is $$3°$$. How far is the person from the monument?

64. 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 $$40°$$. From the same location, the angle of elevation to the top of the antenna is measured to be $$43°$$. Find the height of the antenna.

65. 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 $$36°$$. From the same location, the angle of elevation to the top of the lightning rod is measured to be $$38°$$. Find the height of the lightning rod.

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

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

68. 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.

69. 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.

70. 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 $$60°$$, how far from the base of the tree am I?

61. $$498.3471$$ ft     63. $$1060.09$$ ft     65. $$27.372$$ ft     67. $$22.6506$$ ft     69. $$368.7633$$ ft