Law of Sines: Step-by-Step Guide
The Law of Sines connects the sides of a triangle to the sines of the angles across from them: a / sin(A) = b / sin(B) = c / sin(C). You use it when you know two angles and a side (AAS or ASA), or two sides and an angle that is not between them (SSA, which can be tricky).
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Figure out what you know. Is it AAS, ASA, or SSA? If you know two angles, find the third one (the angles add up to 180 degrees).\[A + B + C = 180^\circ\]
- Set up the proportion. Pair each side with the angle across from it.\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]
- Solve for the unknown. Cross-multiply to get the unknown side or angle by itself.\[x = \frac{a \sin X}{\sin A}\]
- Watch for the ambiguous case (SSA). With SSA, a second angle, 180 degrees minus your first answer, might also work. Check if one or both fit in the triangle.\[B' = 180^\circ - B\]
- Calculate and check. Make sure all the angles add up to 180 degrees and the side lengths make sense.\[A + B + C = 180^\circ\]
Worked example
In triangle ABC, A = 40 deg, B = 60 deg, and a = 8. Find side b.
Step 1: This is AAS. The third angle is C = 180 - 40 - 60 = 80 degrees.
Step 2: a / sin(A) = b / sin(B), so 8 / sin(40 degrees) = b / sin(60 degrees).
Step 3: b = 8 x sin(60 degrees) / sin(40 degrees).
Step 4: There is no ambiguity, because this is AAS.
Step 5: sin(60 degrees) is about 0.8660 and sin(40 degrees) is about 0.6428. So b = 8 x 0.8660 / 0.6428, which is about 10.78.
Check: The angles add up to 180 degrees. Side b is across from the bigger angle (60 degrees), so b should be longer than a (8). It is.
Tips and common mistakes
- Pair each side with the angle across from it. Lowercase a is across from uppercase A.
- If you know two angles, find the third one right away. It gives you more choices.
- For SSA, there may be 0, 1, or 2 triangles. Be careful.
- Use the Law of Sines for AAS, ASA, or SSA. Use the Law of Cosines for SAS or SSS.
- A longer side is always across from a bigger angle. This is a quick way to check.
- Keep your calculator in degree mode if the angles are in degrees.
- When you find an angle, you might get two answers. Throw out any that make the angles add up to more than 180 degrees.
- Keep extra decimal places while you work. Round only at the end.