7. Non-Right Triangles

Use the Law of Sines to find the length of side $a$ to two decimal places.



Use the Law of Sines to find the angle $B$ to the nearest tenth of a degree.



Classify the triangle, then solve: $A=60°,B=15°,c=6$.

An engineer wants to measure the distance to cross a river. If $B=30\degree$, $a=300$$ft$, $C=100\degree$ find the shortest distance (in $ft$) you’d have to travel to cross the river.

Apply the law of sines to the following: a = √5, c = 2√5, A = 30°. What is the value of ​sin​ ​C​​? What is the measure​ of ​​​​C​​​? Based on it​​s angle measures, what kind of triangle is triangl​e ​​ABC​​​?

Fill in the blank(s) to correctly complete each sentence.

A triangle that is not a right triangle is a(n) _________ triangle.

Consider each case and determine whether there is sufficient information to solve the triangle using the law of sines.

Three sides are known.

Which one of the following sets of data does not determine a unique triangle?

a. A = 50°, b = 21, a = 19

b. A = 45°, b = 10, a = 12

c. A = 130°, b = 4, a = 7

d. A = 30°, b = 8, a = 4

Use the law of sines to find the indicated part of each triangle ABC.

Find b if C = 74.2°, c = 96.3 m, B = 39.5

Find each angle B. Do not use a calculator.

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Find the length of each side labeled a. Do not use a calculator.

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Find the unknown angles in triangle ABC for each triangle that exists.

A = 29.7°, b = 41.5 ft, a = 27.2 ft

Solve each triangle ABC.

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Find the unknown angles in triangle ABC for each triangle that exists.

C = 41° 20', b = 25.9 m, c = 38.4 m

Solve each triangle ABC.

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Solve each triangle ABC.

A = 68.41°, B = 54.23°, a = 12.75 ft

Find the unknown angles in triangle ABC for each triangle that exists.

B = 74.3°, a = 859 m, b = 783 m

Find the unknown angles in triangle ABC for each triangle that exists.

A = 142.13°, b = 5.432 ft, a = 7.297 ft

Solve each triangle ABC.

B = 38° 40', a = 19.7 cm, C = 91° 40'

Solve each triangle ABC that exists.

A = 42.5°, a = 15.6 ft, b = 8.14 ft

Solve each triangle ABC.

A = 39.70°, C = 30.35°, b = 39.74 m

Solve each triangle ABC.

B = 42.88°, C = 102.40°, b = 3974 ft

Solve each triangle ABC that exists.

B = 72.2°, b = 78.3 m, c = 145 m

Solve each triangle ABC that exists.

A = 38° 40', a = 9.72 m, b = 11.8 m

Solve each triangle ABC that exists.

A = 96.80°, b = 3.589 ft, a = 5.818 ft

Solve each triangle ABC.

C = 79° 18', c = 39.81 mm, A = 32° 57'

Solve each triangle ABC that exists.

B = 39.68°, a = 29.81 m, b = 23.76 m

Use the law of sines to find the indicated part of each triangle ABC.

Find B if C = 51.3°, c = 68.3 m, b = 58.2 m

To find the distance AB across a river, a surveyor laid off a distance BC = 354 m on one side of the river. It is found that B = 112° 10' and C = 15° 20'. Find AB. See the figure.

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To determine the distance RS across a deep canyon, Rhonda lays off a distance TR = 582 yd​​​​. She then finds that T = 32° 50' and R = 102° 20'. Find ​​RS​​. See the figure.

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A ship is sailing due north. At a certain point the bearing of a lighthouse 12.5 km away is N 38.8° E. Later on, the captain notices that the bearing of the lighthouse has become S 44.2° E. How far did the ship travel between the two observations of the lighthouse?

Radio direction finders are placed at points A and B, which are 3.46 mi apart on an east-west line, with A west of B. From A the bearing of a certain radio transmitter is 47.7°, and from B the bearing is 302.5°. Find the distance of the transmitter from A.

The bearing of a lighthouse from a ship was found to be N 37° E. After the ship sailed 2.5 mi due south, the new bearing was N 25° E. Find the distance between the ship and the lighthouse at each location.

Standing on one bank of a river flowing north, Mark notices a tree on the opposite bank at a bearing of 115.45°. Lisa is on the same bank as Mark, but 428.3 m away. She notices that the bearing of the tree is 45.47°. The two banks are parallel. What is the distance across the river?

Use the law of sines to find the indicated part of each triangle ABC.

Find b if a = 165 m, A = 100.2°, B = 25.0°

Find the area of each triangle using the formula 𝓐 = ½ bh, and then verify that the formula 𝓐 = ½ ab sin C gives the same result.

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Determine the number of triangles ABC possible with the given parts.

a = 50, b = 26, A = 95°

Find the area of each triangle ABC.

A = 42.5°, b = 13.6 m, c = 10.1 m

Find the area of each triangle ABC.

B = 124.5°, a = 30.4 cm, c = 28.4 cm

Find the area of each triangle ABC.

A = 56.80°, b = 32.67 in., c = 52.89 in.

Find the area of each triangle ABC.

A = 59.80°, b = 15.00 cm, C = 53.10°

A painter is going to apply paint to a triangular metal plate on a new building. Two sides measure 16.1 m and 15.2 m, and the angle between the sides is 125°. What is the area of the surface to be painted?

​A real estate agent wants to find the area of a triangular lot. A surveyor takes measurements and finds that two sides are 52.1 m and 21.3 m, and the angle between them is 42.2°. What is the area of the triangular lot?

Use the law of sines to find the indicated part of each triangle ABC.

Find A if C = 79° 20', c = 97.4 mm, a = 75.3 mm

Determine the number of triangles ABC possible with the given parts.

a = 31, b = 26, B = 48°

Determine the number of triangles ABC possible with the given parts.

c = 50, b = 61, C = 58°

In each figure, a line segment of length L is to be drawn from the given point to the positive x-axis in order to form a triangle. For what value(s) of L can we draw the following?

a. two triangles

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In each figure, a line segment of length L is to be drawn from the given point to the positive x-axis in order to form a triangle. For what value(s) of L can we draw the following?

b. exactly one triangle

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In each figure, a line segment of length L is to be drawn from the given point to the positive x-axis in order to form a triangle. For what value(s) of L can we draw the following?

c. no triangle

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Without using the law of sines, explain why no triangle ABC can exist that satisfies A = 103° 20', a = 14.6 ft, b = 20.4 ft.

Apply the law of sines to the following:

A = 104°, a = 26.8, b = 31.3.

What happens when we try to find the measure of angle B using a calculator?

Use the law of sines to prove that each statement is true for any triangle ABC, with corresponding sides a, b, and c.

(a - b)/(a + b) = (sin A - sin B)/(sin A + sin B)

A balloonist is directly above a straight road 1.5 mi long that joins two villages. She finds that the town closer to her is at an angle of depression of 35°, and the farther town is at an angle of depression of 31°. How high above the ground is the balloon? 

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Use the law of sines to find the indicated part of each triangle ABC.

Find A if B = 39° 50', b = 268 m, a = 340 m