Triangle Trigonometry: Get Stronger

Right Triangle Trigonometry

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

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

For the following exercises, use cofunctions of complementary angles.

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

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

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

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

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

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

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

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

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

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

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

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

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

27. [latex]\sec 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.

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.

33.
A right triangle with sides of 7, b, and c. Angles of 35 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.

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

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

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

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.

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.

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?

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?

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.

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?

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.

Inverse Trigonometric Functions

1. Why do the functions [latex]f(x)=\sin^{−1}x[/latex] and [latex]g(x)=\cos^{−1}x[/latex] have different ranges?

4. Most calculators do not have a key to evaluate [latex]\sec^{−1}(2)[/latex]. Explain how this can be done using the cosine function or the inverse cosine function.

For the following exercises, evaluate the expressions.

9. [latex]\sin^{−1}\left(−\frac{1}{2}\right)[/latex]

11. [latex]\cos^{−1}\left(−\frac{\sqrt{2}}{2}\right)[/latex]

15. [latex]\tan^{−1}(\sqrt{3})[/latex]

For the following exercises, use a calculator to evaluate each expression. Express answers to the nearest hundredth.

17. [latex]\cos^{−1}(−0.4)[/latex]

19. [latex]\arccos\left(\frac{3}{5}\right)[/latex]

21. [latex]\tan^{−1}(6)[/latex]

For the following exercises, find the angle θ in the given right triangle. Round answers to the nearest hundredth.

23.
An illustration of a right triangle with angle theta. Adjacent the angle theta is a side of length 19. Opposite the angle theta is a side with length 12.

For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why.

25. [latex]\tan^{−1}(\sin(\pi))[/latex]

29. [latex]\tan^{−1}\left(\sin\left(\frac{4\pi}{3}\right)\right)[/latex]

31. [latex]\tan^{−1}\left(\sin\left(\frac{−5\pi}{2}\right)\right)[/latex]

33. [latex]\sin\left(\cos^{−1}\left(\frac{3}{5}\right)\right)[/latex]

35. [latex]\cos\left(\tan^{−1}\left(\frac{12}{5}\right)\right)[/latex]

For the following exercises, find the function if [latex]\sin t=\frac{x}{x+1}[/latex].

43. [latex]\cos t[/latex]

45. [latex]\cot t[/latex]

47. [latex]\tan^{−1}\left({x}{\sqrt{2x+1}}\right)[/latex]

49. Graph [latex]y=\arccos x[/latex] and state the domain and range of the function.

53. Suppose a 13-foot ladder is leaning against a building, reaching to the bottom of a second-floor window 12 feet above the ground. What angle, in radians, does the ladder make with the building?

55. An isosceles triangle has two congruent sides of length 9 inches. The remaining side has a length of 8 inches. Find the angle that a side of 9 inches makes with the 8-inch side.

57. A truss for the roof of a house is constructed from two identical right triangles. Each has a base of 12 feet and height of 4 feet. Find the measure of the acute angle adjacent to the 4-foot side.

61. A 20-foot ladder leans up against the side of a building so that the foot of the ladder is 10 feet from the base of the building. If specifications call for the ladder’s angle of elevation to be between 35 and 45 degrees, does the placement of this ladder satisfy safety specifications?

Non-right Triangles: Law of Sines

3. When can you use the Law of Sines to find a missing angle?5. What type of triangle results in an ambiguous case?

For the following exercises, assume [latex]\alpha[/latex] is opposite side [latex]a,\beta[/latex] is opposite side [latex]b[/latex], and [latex]\gamma[/latex] is opposite side [latex]c[/latex]. Solve each triangle, if possible. Round each answer to the nearest tenth.

7. [latex]\alpha =35^\circ ,\gamma =73^\circ ,c=20[/latex]

9. [latex]a=4,\alpha =60^\circ ,\beta =100^\circ[/latex]

For the following exercises, use the Law of Sines to solve for the missing side for each oblique triangle. Round each answer to the nearest hundredth. Assume that angle [latex]A[/latex] is opposite side [latex]a[/latex], angle [latex]B[/latex] is opposite side [latex]b[/latex], and angle [latex]C[/latex] is opposite side [latex]c[/latex].

11. Find side [latex]b[/latex] when [latex]A=37^\circ ,B=49^\circ ,c=5[/latex].

13. Find side [latex]c[/latex] when [latex]B=37^\circ ,C=21,b=23[/latex].

For the following exercises, assume [latex]\alpha[/latex] is opposite side [latex]a,\beta[/latex] is opposite side [latex]b[/latex], and [latex]\gamma[/latex] is opposite side [latex]c[/latex]. Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Round each answer to the nearest tenth.

15. [latex]\gamma =113^\circ ,b=10,c=32[/latex]

17. [latex]a=12,c=17,\alpha =35^\circ[/latex]

19. [latex]a=7,c=9,\alpha =43^\circ[/latex]

21. [latex]b=13,c=5,\gamma =10^\circ[/latex]

23. [latex]\beta =119^\circ ,b=8.2,a=11.3[/latex]

For the following exercise, use the Law of Sines to solve, if possible, the missing side or angle for each triangle or triangles in the ambiguous case. Round each answer to the nearest tenth.

25. Find angle [latex]A[/latex] when [latex]a=13,b=6,B=20^\circ[/latex].

For the following exercises, find the area of the triangle with the given measurements. Round each answer to the nearest tenth.

27. [latex]a=5,c=6,\beta =35^\circ[/latex]

29. [latex]a=32,b=24,\gamma =75^\circ[/latex]

For the following exercises, find the length of side [latex]x[/latex]. Round to the nearest tenth.

31.
A triangle with an angle of 50 degrees and opposite side of length 10. Another angle is 70 degrees with side opposite of length x.

33.
A triangle. One angle is 45 degrees with side opposite = x. Another angle is 75 degrees. The side adjacent to the 45 and 75 degree angles = 15.

35.
A triangle. One angle is 50 degrees with opposite = x. Another angle is 42 degrees with opposite side = 14.

For the following exercises, find the measure of angle [latex]x[/latex], if possible. Round to the nearest tenth.

37.
A triangle. One angles is 98 degrees with opposite side = 10. Another angle is x degrees with opposite side = 5.

39.
A triangle. One angle is 22 degrees with side opposite = 5. Another angle is x degrees with opposite side = 13.

41. Notice that [latex]x[/latex] is an obtuse angle.
A triangle. One angle is 55 degrees with side opposite = 21. Another angle is x degrees with opposite side = 24.

For the following exercises, find the area of each triangle. Round each answer to the nearest tenth.

43.
A triangle. One angle is 93 degrees with opposite side = 32.6. Another side is 24.1.

45.
A triangle. One angle is 25 degrees. The two sides adjacent to that angle are 18 and 15

47.
A triangle. One angle is 58 degrees with opposite side unknown. Another angle is 51 degrees with opposite side = 9. The side adjacent to the two given angles is 11.

49.
A triangle. One angle is 115 degrees with opposite side = 50. Another angle is 30 degrees with opposite side = 30.

59. A pole leans away from the sun at an angle of [latex]7^\circ[/latex] to the vertical, as shown in below. When the elevation of the sun is [latex]55^\circ[/latex], the pole casts a shadow 42 feet long on the level ground. How long is the pole? Round the answer to the nearest tenth.
A triangle within a triangle. The outer triangle is formed by vertices A, B, and S (the sun). Side A B is the horizontal base, the ground, and is 42 feet. Angle A is 55 degrees. The inner triangle is formed by vertices A, B, and C. Side B C is the pole. Vertex C is located on side A S of the outer triangle between vertices A and S. Angle C B S is 7 degrees.

61. The diagram below shows a satellite orbiting Earth. The satellite passes directly over two tracking stations [latex]A[/latex] and [latex]B[/latex], which are 69 miles apart. When the satellite is on one side of the two stations, the angles of elevation at [latex]A[/latex] and [latex]B[/latex] are measured to be [latex]86.2^\circ[/latex] and [latex]83.9^\circ[/latex], respectively. How far is the satellite from station [latex]A[/latex] and how high is the satellite above the ground? Round answers to the nearest whole mile.
A triangle formed by two ground tracking stations A and B and the satellite. Side A B is the horizontal base of the triangle. Angle A is 83.9 degrees, and the supplementary angle to angle B is 86.2 degrees.

63. The roof of a house is at a [latex]20^\circ[/latex] angle. An 8-foot solar panel is to be mounted on the roof and should be angled [latex]38^\circ[/latex] relative to the horizontal for optimal results. How long does the vertical support holding up the back of the panel need to be? Round to the nearest tenth.
A triangle whose sides are the solar panel, the roof which goes past the solar panel, and the vertical support for the panel. The solar panel side is 8 feet long. There are horizontal dotted lines at the bottom of the solar panel and the bottom of the roof. The angle between the solar panel and the horizontal is 38 degrees. The angle between the roof and the horizontal is 20 degrees.

65. A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 4.3 km apart, to be 32° and 56°, as shown below. Find the distance of the plane from point [latex]A[/latex] to the nearest tenth of a kilometer.
A triangle formed between the plane and two points on the ground, A and B. Side A B is the horizontal base. The plane is above and to the left of both A and B. Point B is to the right of point A. There is a dotted horizontal line going through the plane parallel to the ground. The angle formed between point B, the plane, and the dotted horizontal line is 32 degrees. The angle formed between point A, the plane, and the dotted horizontal line is 56 degrees.

67. In order to estimate the height of a building, two students stand at a certain distance from the building at street level. From this point, they find the angle of elevation from the street to the top of the building to be 35°. They then move 250 feet closer to the building and find the angle of elevation to be 53°. Assuming that the street is level, estimate the height of the building to the nearest foot.

71. A street light is mounted on a pole. A 6-foot-tall man is standing on the street a short distance from the pole, casting a shadow. The angle of elevation from the tip of the man’s shadow to the top of his head of 28°. A 6-foot-tall woman is standing on the same street on the opposite side of the pole from the man. The angle of elevation from the tip of her shadow to the top of her head is 28°. If the man and woman are 20 feet apart, how far is the street light from the tip of the shadow of each person? Round the distance to the nearest tenth of a foot.

73. Two streets meet at an 80° angle. At the corner, a park is being built in the shape of a triangle. Find the area of the park if, along one road, the park measures 180 feet, and along the other road, the park measures 215 feet.

75. The Bermuda triangle is a region of the Atlantic Ocean that connects Bermuda, Florida, and Puerto Rico. Find the area of the Bermuda triangle if the distance from Florida to Bermuda is 1030 miles, the distance from Puerto Rico to Bermuda is 980 miles, and the angle created by the two distances is 62°.

Non-right Triangles: Law of Cosines

1. If you are looking for a missing side of a triangle, what do you need to know when using the Law of Cosines?2. If you are looking for a missing angle of a triangle, what do you need to know when using the Law of Cosines?3. Explain what [latex]s[/latex] represents in Heron’s formula.

For the following exercises, assume [latex]\alpha[/latex] is opposite side [latex]a,\beta[/latex] is opposite side [latex]b[/latex], and [latex]\gamma[/latex] is opposite side [latex]c[/latex]. If possible, solve each triangle for the unknown side. Round to the nearest tenth.

7. [latex]\alpha =120^\circ ,b=6,c=7[/latex]

13. [latex]\alpha =43.1^\circ ,a=184.2,b=242.8[/latex]

15. [latex]\beta =50^\circ ,a=105,b=45{}_{}{}^{}[/latex]

For the following exercises, use the Law of Cosines to solve for the missing angle of the oblique triangle. Round to the nearest tenth.

17. [latex]a=14,\text{ }b=13,\text{ }c=20[/latex]; find angle [latex]C[/latex].

19. [latex]a=13,b=22,c=28[/latex]; find angle [latex]A[/latex].

For the following exercises, solve the triangle. Round to the nearest tenth.

21. [latex]A=35^\circ ,b=8,c=11[/latex]

23. [latex]C=121^\circ ,a=21,b=37[/latex]

25. [latex]a=3.1,b=3.5,c=5[/latex]

For the following exercises, use Heron’s formula to find the area of the triangle. Round to the nearest hundredth.

27. Find the area of a triangle with sides of length 18 in, 21 in, and 32 in. Round to the nearest tenth.

29. [latex]a=\frac{1}{2}\text{m},b=\frac{1}{3}\text{m},c=\frac{1}{4}\text{m}[/latex]

31. [latex]a=1.6\text{ yd},\text{ }b=2.6\text{ yd},\text{ }c=4.1\text{ yd}[/latex]

For the following exercises, find the length of side [latex]x[/latex]. Round to the nearest tenth.

33.
A triangle. One angle is 42 degrees with opposite side = x. The other two sides are 4.5 and 3.4.Ω

37.
A triangle. One angle is 123 degrees with opposite side = x. The other two sides are 1/5 and 1/3.

For the following exercises, find the measurement of angle [latex]A[/latex].

39.
A triangle. Angle A is opposite a side of length 125. The other two sides are 115 and 100.

41.
A triangle. Angle A is opposite a side of length 40.6. The other two sides are 38.7 and 23.3.

For the following exercises, solve for the unknown side. Round to the nearest tenth.

43.
A triangle. One angle is 60 degrees with opposite side unknown. The other two sides are 20 and 28.

For the following exercises, find the area of the triangle. Round to the nearest hundredth.

47.
A triangle with sides 8, 12, and 17. Angles unknown.

49.
A triangle with sides 1.9, 2.6, and 4.3. Angles unknown.

51.
A triangle with sides 1/2, 2/3, and 3/5. Angles unknown.

For the following exercises, find the area of the triangle.

59.
A triangle. One angle is 22 degrees with opposite side = 3.4. Another side is 5.3.

61.
A triangle. One angle is 18 degrees with opposite side = 12.8. Another side is 18.8.

63. A satellite calculates the distances and angle shown in the diagram below (not to scale). Find the distance between the two cities. Round answers to the nearest tenth.
Insert figure(table) alt text: A triangle formed by two cities on the ground and a satellite above them. The angle by the satellite is 2.1 degrees with opposite side unknown, which is the distance between the two cities. The lengths of the other sides are 370 and 350 km.

65. A 113-foot tower is located on a hill that is inclined 34° to the horizontal, as shown in the image below. A guy-wire is to be attached to the top of the tower and anchored at a point 98 feet uphill from the base of the tower. Find the length of wire needed.
Two triangles, one on top of the other. The bottom triangle is the hill inclined 34 degrees to the horizontal. The second is formed by the base of the tower on the incline of the hill, the top of the tower, and the wire anchor point uphill from the tower on the incline. The sides are the tower, the incline of the hill, and the wire. The tower side is 113 feet and the incline side is 98 feet.

67. The graph in the figure below represents two boats departing at the same time from the same dock. The first boat is traveling at 18 miles per hour at a heading of 327° and the second boat is traveling at 4 miles per hour at a heading of 60°. Find the distance between the two boats after 2 hours.
Insert figure(table) alt text: A graph of two rays, which represent the paths of the two boats. Both rays start at the origin. The first goes into the first quadrant at a 60 degree angle at 4 mph. The second goes into the fourth quadrant at a 327 degree angle from the origin. The second travels at 18 mph.

69. A pilot flies in a straight path for 1 hour 30 min. She then makes a course correction, heading 10° to the right of her original course, and flies 2 hours in the new direction. If she maintains a constant speed of 680 miles per hour, how far is she from her starting position?

71. Philadelphia is 140 miles from Washington, D.C., Washington, D.C. is 442 miles from Boston, and Boston is 315 miles from Philadelphia. Draw a triangle connecting these three cities and find the angles in the triangle.

73. Two airplanes take off in different directions. One travels 300 mph due west and the other travels 25° north of west at 420 mph. After 90 minutes, how far apart are they, assuming they are flying at the same altitude?

75. The four sequential sides of a quadrilateral have lengths 4.5 cm, 7.9 cm, 9.4 cm, and 12.9 cm. The angle between the two smallest sides is 117°. What is the area of this quadrilateral?

77. Find the area of a triangular piece of land that measures 30 feet on one side and 42 feet on another; the included angle measures 132°. Round to the nearest whole square foot.