Section 5.1 – Non-Right Triangles: Law of Sines
Learning Objectives
Welcome to Section 5.1! In this section, you will:
- Use the Law of Sines to solve oblique triangles.
- Find the area of an oblique triangle using the sine function.
- Solve applied problems using the Law of Sines.
To ensure the safety of over 5,000 U.S. aircraft flying simultaneously during peak times, air traffic controllers monitor and communicate with them after receiving data from the robust radar beacon system. Suppose two radar stations located 20 miles apart each detect an aircraft between them. The angle of elevation measured by the first station is 35 degrees, whereas the angle of elevation measured by the second station is 15 degrees. How can we determine the altitude of the aircraft? We see in Figure 1 that the triangle formed by the aircraft and the two stations is not a right triangle, so we cannot use what we know about right triangles. In this section, we will find out how to solve problems involving non-right triangles.

Using the Law of Sines to Solve Oblique Triangles
In any triangle, we can draw an altitude, a perpendicular line from one vertex to the opposite side, forming two right triangles. It would be preferable, however, to have methods that we can apply directly to non-right triangles without first having to create right triangles.
Any triangle that is not a right triangle is an oblique triangle. Solving an oblique triangle means finding the measurements of all three angles and all three sides. To do so, we need to start with at least three of these values, including at least one of the sides. We will investigate three possible oblique triangle problem situations:
- ASA (angle-side-angle) We know the measurements of two angles and the included side. See Figure 2.
Figure 2 - AAS (angle-angle-side) We know the measurements of two angles and a side that is not between the known angles. See Figure 3.
Figure 3 - SSA (side-side-angle) We know the measurements of two sides and an angle that is not between the known sides. See Figure 4.

Knowing how to approach each of these situations enables us to solve oblique triangles without having to drop a perpendicular to form two right triangles. Instead, we can use the fact that the ratio of the measurement of one of the angles to the length of its opposite side will be equal to the other two ratios of angle measure to opposite side. Let’s see how this statement is derived by considering the triangle shown in Figure 5.

Using the right triangle relationships, we know that
We then set the expressions equal to each other.
Similarly, we can compare the other ratios.
Collectively, these relationships are called the Law of Sines.
Note the standard way of labeling triangles: angle
While calculating angles and sides, be sure to carry the exact values through to the final answer. Generally, final answers are rounded to the nearest tenth, unless otherwise specified.

Law of Sines
Given a triangle with angles and opposite sides labeled as in Figure 6, the ratio of the measurement of an angle to the length of its opposite side will be equal to the other two ratios of angle measure to opposite side. All proportions will be equal. The Law of Sines is based on proportions and is presented symbolically two ways.
To solve an oblique triangle, use any pair of applicable ratios.
Example 1
Solving for Two Unknown Sides and Angle of an AAS Triangle
Solve the triangle shown in Figure 7 to the nearest tenth.

Show/Hide Solution
The three angles must add up to 180 degrees. From this, we can determine that
To find an unknown side, we need to know the corresponding angle and a known ratio. We know that angle
Similarly, to solve for
Therefore, the complete set of angles and sides is
Using The Law of Sines to Solve SSA Triangles
We can use the Law of Sines to solve any oblique triangle, but some solutions may not be straightforward. In some cases, more than one triangle may satisfy the given criteria, which we describe as an ambiguous case. Triangles classified as SSA, those in which we know the lengths of two sides and the measurement of the angle opposite one of the given sides, may result in one or two solutions, or even no solution.
Possible Outcomes for SSA Triangles
Oblique triangles in the category SSA may have four different outcomes. Figure 9 illustrates the solutions with the known sides

Example 2
Solving an Oblique SSA Triangle
Solve the triangle in Figure 10 for the missing side and find the missing angle measures to the nearest tenth.

Show/Hide Solution
Use the Law of Sines to find angle
However, in the diagram, angle

The angle supplementary to
We can then use these measurements to solve the other triangle. Since
Now we need to find
We have
Finally,
To summarize, there are two triangles with an angle of 35°, an adjacent side of 8, and an opposite side of 6, as shown in Figure 12.

However, we were looking for the values for the triangle with an obtuse angle
Try It #2
Given
Example 3
Solving for the Unknown Sides and Angles of a SSA Triangle
In the triangle shown in Figure 13, solve for the unknown side and angles. Round your answers to the nearest tenth.

Show/Hide Solution
In choosing the pair of ratios from the Law of Sines to use, look at the information given. In this case, we know the angle
To find
In this case, if we subtract
which is impossible, and so
To find the remaining missing values, we calculate
The complete set of solutions for the given triangle is
Try It #3
Given
Example 4
Finding the Triangles That Meet the Given Criteria
Find all possible triangles if one side has length 4 opposite an angle of 50°, and a second side has length 10.
Show/Hide Solution
Using the given information, we can solve for the angle opposite the side of length 10. See Figure 14.

We can stop here without finding the value of
Try It #4
Determine the number of triangles possible given
Finding the Area of an Oblique Triangle Using the Sine Function
Now that we can solve a triangle for missing values, we can use some of those values and the sine function to find the area of an oblique triangle. Recall that the area formula for a triangle is given as

Thus,
Similarly,
Area of an Oblique Triangle
The formula for the area of an oblique triangle is given by
This is equivalent to one-half of the product of two sides and the sine of their included angle.
Example 5
Finding the Area of an Oblique Triangle
Find the area of a triangle with sides
Show/Hide Solution
Using the formula, we have
Try It #5
Find the area of the triangle given
Solving Applied Problems Using the Law of Sines
The more we study trigonometric applications, the more we discover that the applications are countless. Some are flat, diagram-type situations, but many applications in calculus, engineering, and physics involve three dimensions and motion.
Example 6
Finding an Altitude
Find the altitude of the aircraft in the problem introduced at the beginning of this section, shown in Figure 16. Round the altitude to the nearest tenth of a mile.

Show/Hide Solution
To find the elevation of the aircraft, we first find the distance from one station to the aircraft, such as the side
Because the angles in the triangle add up to 180 degrees, the unknown angle must be 180°−15°−35°=130°. This angle is opposite the side of length 20, allowing us to set up a Law of Sines relationship.
The distance from one station to the aircraft is about 14.98 miles.
Now that we know
The aircraft is at an altitude of approximately 3.9 miles.
Try It #6
The diagram shown in Figure 17 represents the height of a blimp flying over a football stadium. Find the height of the blimp if the angle of elevation at the southern end zone, point A, is 70°, the angle of elevation from the northern end zone, point

Example 7
Finding the length of a ski lift
A ski resort in Red River, New Mexico wants to build a new ski lift up the side of the mountain. A surveyor measures
Show/Hide Solution
To use the Laws of Sines, we need to know one more angle and side pair of the triangle PRQ. The length of PR is known. Therefore, to find the length of the ski lift, we first find the angle opposite to PR. Because
We can now use the Laws of Sines to find the length of the ski lift.
The length of the ski lift is approximately 1490.48 feet.
Example 8
Finding height of a distance object
A tree stand on a hillside of slope 28° (from the horizontal). From a point 75 feet down the hill, the angle of elevation to the top of the tree is 45°. What is the height of the tree?
Show/Hide Solution
To use the Laws of Sines, we need to know the angle opposite to the side of interest and one more angle and side pair. The angle opposite to the side of interest is 45°-29°=17°. The length of the hill is known. Therefore, to find the height of the tree, we first find the angle opposite to the length of the hill. If we draw a horizontal line at the top of the tree parallel to the ground, the angle that is compliment to the angle opposite to the length of the hill and the angle of elevation to the top of the tree are alternate interior angles, therefore they are both 45°.
Because complementary angles add up to 90 degrees, the angle opposite to the length of the hill must be 90°−45°=45°. We can now use the Laws of Sines to find the height of the tree.
The height of the tree is approximately 31.01 feet.
Media
Access these online resources for additional instruction and practice with trigonometric applications.
Law of Sines: The Ambiguous Case
5.1 Section Exercises
Verbal
1. Describe the altitude of a triangle.
2. Compare right triangles and oblique triangles.
3. When can you use the Law of Sines to find a missing angle?
4. In the Law of Sines, what is the relationship between the angle in the numerator and the side in the denominator?
5. What type of triangle results in an ambiguous case?
Algebraic
For the following exercises, assume
6.
7.
8.
9.
10.
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
11. Find side
12. Find side
13. Find side
For the following exercises, assume
14.
15.
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19.
20.
21.
22.
23.
For the following exercises, 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.
24. Find angle
25. Find angle
26. Find angle
For the following exercises, find the area of the triangle with the given measurements. Round each answer to the nearest tenth.
27.
28.
29.
30.
Graphical
For the following exercises, find the length of side
31.
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34.
35.
36.
For the following exercises, find the measure of angle
37.
38.
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40.
41. Notice that
42.
For the following exercise, solve the triangle. Round each answer to the nearest tenth.
43.
44. For the following exercises, find the area of each triangle. Round each answer to the nearest tenth.
45.
46.
47.
48.
49.
Extensions
50.
Find the radius of the circle in Figure 18. Round to the nearest tenth.

51. Find the diameter of the circle in Figure 19. Round to the nearest tenth.

52. Find

53. Find

54. Solve both triangles in Figure 22. Round each answer to the nearest tenth.

55. Find

56. Solve the triangle in Figure 24. (Hint: Draw a perpendicular from

57. Solve the triangle in Figure 25. (Hint: Draw a perpendicular from

58. In Figure 26,

Real-World Applications
59.
A pole leans away from the sun at an angle of

60. To determine how far a boat is from shore, two radar stations 500 feet apart find the angles out to the boat, as shown in Figure 28. Determine the distance of the boat from station

61. Figure 29 shows a satellite orbiting Earth. The satellite passes directly over two tracking stations

62. A communications tower is located at the top of a steep hill, as shown in Figure 30. The angle of inclination of the hill is

63. The roof of a house is at a

64. Similar to an angle of elevation, an angle of depression is the acute angle formed by a horizontal line and an observer’s line of sight to an object below the horizontal. A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 6.6 km apart, to be

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 in Figure 33. Find the distance of the plane from point

66. 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 39°. They then move 300 feet closer to the building and find the angle of elevation to be 50°. Assuming that the street is level, estimate the height of the building to the nearest foot.
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.
68. Points
69. A man and a woman standing
70. Two search teams spot a stranded climber on a mountain. The first search team is 0.5 miles from the second search team, and both teams are at an altitude of 1 mile. The angle of elevation from the first search team to the stranded climber is 15°. The angle of elevation from the second search team to the climber is 22°. What is the altitude of the climber? Round to the nearest tenth of a mile.
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.
72. Three cities,
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.
74. Brian’s house is on a corner lot. Find the area of the front yard if the edges measure 40 and 56 feet, as shown in Figure 34.

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°.
76. A yield sign measures 30 inches on all three sides. What is the area of the sign?
77. Naomi bought a dining table whose top is in the shape of a triangle. Find the area of the table top if two of the sides measure 4 feet and 4.5 feet, and the smaller angles measure 32° and 42°, as shown in Figure 35.
