Calculate a missing side or angle of any triangle from two angles and a side, or from two sides and a non-included angle, using the law of sines.
Law of Sines Formula
The law of sines states that in any triangle the ratio of a side length to the sine of its opposite angle is the same for all three sides:
a / sin(A) = b / sin(B) = c / sin(C)
To solve for an unknown side when you know its opposite angle and one complete side-angle pair, rearrange the ratio:
a = b * sin(A) / sin(B)
To solve for an unknown angle when you know its opposite side and one complete side-angle pair, use:
sin(A) = a * sin(B) / b
- a, b, c: the lengths of the three sides of the triangle
- A, B, C: the angles opposite sides a, b, and c, measured in degrees or radians
- sin: the sine function applied to each angle
The calculator works from one known side-angle pair, since that pair fixes the value of the common ratio a / sin(A). In the "two angles and one side" mode (ASA or AAS) it first finds the third angle from A + B + C = 180 degrees, then applies the ratio to find the two missing sides. In the "two sides and a non-included angle" mode (SSA) it solves for the unknown angle first using the sine relationship, then completes the triangle. You can also use the ratio mode to read the shared value a / sin(A) directly, or to find a single side or angle from that ratio.
Triangle Cases and What Each Produces
The number of valid triangles you can build depends on which pieces you are given. The SSA case is called the ambiguous case because it can yield zero, one, or two triangles.
| Given | Case | Possible triangles |
|---|---|---|
| Two angles and the included side | ASA | Exactly one |
| Two angles and a non-included side | AAS | Exactly one |
| Two sides and an angle opposite one of them | SSA | Zero, one, or two |
For the SSA case, the table below shows how to read the result once you compare the side opposite the known angle to the height h = (other side) * sin(known angle).
| Condition (known angle A, sides a and b) | Number of triangles |
|---|---|
| a is less than b * sin(A) | Zero |
| a equals b * sin(A) | One (right triangle) |
| b * sin(A) is less than a, and a is less than b | Two |
| a is greater than or equal to b | One |
Example Problems
Example 1 (AAS, find a side). You know angle A = 40 degrees, angle B = 60 degrees, and side a = 8. First find side b using the ratio. Apply b = a * sin(B) / sin(A) = 8 * sin(60) / sin(40) = 8 * 0.8660 / 0.6428 = 10.78. Side b is about 10.78.
Example 2 (SSA, find an angle). You know side a = 7, side b = 9, and angle A = 35 degrees. Solve sin(B) = b * sin(A) / a = 9 * sin(35) / 7 = 9 * 0.5736 / 7 = 0.7375. Then B = arcsin(0.7375) = 47.5 degrees. Because this is the SSA case, also check the supplement 180 - 47.5 = 132.5 degrees; since 35 + 132.5 = 167.5 is still less than 180, a second triangle exists, so this problem has two solutions.
Frequently Asked Questions
When should I use the law of sines instead of the law of cosines? Use the law of sines when you know a side and the angle opposite it, plus one more side or angle. That covers the ASA, AAS, and SSA cases. Use the law of cosines instead when you know three sides (SSS) or two sides and the included angle (SAS), because in those cases you do not yet have a matched side-angle pair to start the sine ratio.
What is the ambiguous case? The ambiguous case is the SSA situation, where you are given two sides and an angle opposite one of them. Depending on the numbers, this can describe zero, one, or two different triangles. When you solve for the unknown angle with arcsine, there can be two angles between 0 and 180 degrees with the same sine, so you should check whether the supplementary angle also produces a valid triangle whose angles still sum to 180 degrees.
Can I enter angles in radians? Yes. Choose the angle unit before entering values. If you select radians, type each angle as a radian measure, and any angle results are returned in the same unit. The internal ratio is identical either way, since the calculator applies the sine function to whatever unit you select.
