Trigonometry · Quick Reference

Law of Sines Cheat Sheet

Trigonometry · Lesson 6/8

In one line: the Law of Sines connects the sides and angles of ANY triangle, not just right triangles — a powerful tool for solving triangles when you don't have a 90° angle to work with.

Key Ideas

1The Law of Sines Formula. a/sin(A) = b/sin(B) = c/sin(C), where lowercase letters are side lengths and uppercase letters are the angles opposite those sides.

2When to Use It. Use the Law of Sines when you know: two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA).

3Solving for a Missing Side. Set up a proportion using the Law of Sines, then cross-multiply and solve.

4Solving for a Missing Angle. Rearrange the Law of Sines to solve for the unknown angle using inverse sine.

5The Ambiguous Case (SSA). When given two sides and a non-included angle, there may be zero, one, or two possible triangles — always worth checking carefully.

Worked Examples

In a triangle, angle A=30°, angle B=70°, side a=10. Find side b.

b is approximately 18.8

In a triangle, angle A=40°, side a=8, side b=10. Find angle B.

B is approximately 53.5°

In a triangle, angle A=90°, angle B=45°. Find angle C.

C = 45°

Formulas

a/sin(A) = b/sin(B) = c/sin(C)

Practice Yourself

A triangle has angle A=45°, angle B=45°, side a=10. Find side b.

b=10 (since angle A = angle B, sides are equal)

A triangle has angle A=30°, side a=5, side b=8. Set up the Law of Sines proportion to find angle B (don't solve).

5/sin(30°) = 8/sin(B)

What information do you need to apply the Law of Sines?

A matching angle-side pair, plus one more angle or side