01 Key Concepts
The 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.
When 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).
Solving for a Missing Side
Set up a proportion using the Law of Sines, then cross-multiply and solve.
Solving for a Missing Angle
Rearrange the Law of Sines to solve for the unknown angle using inverse sine.
The 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.
02 Key Formulas
- a/sin(A) = b/sin(B) = c/sin(C)
03 Solved Examples
Example 1 In a triangle, angle A=30°, angle B=70°, side a=10. Find side b.
- Set up the proportion: a/sin(A) = b/sin(B).
- 10/sin(30°) = b/sin(70°).
- 10/0.5 = b/0.94.
- 20 = b/0.94, so b = 20*0.94.
Answer: b is approximately 18.8
Example 2 In a triangle, angle A=40°, side a=8, side b=10. Find angle B.
- Set up the proportion: a/sin(A) = b/sin(B).
- 8/sin(40°) = 10/sin(B).
- 8/0.643 = 10/sin(B).
- 12.44 = 10/sin(B), so sin(B) = 10/12.44 = 0.804.
- B = arcsin(0.804).
Answer: B is approximately 53.5°
Example 3 In a triangle, angle A=90°, angle B=45°. Find angle C.
- All three angles sum to 180°.
- 180 - 90 - 45.
Answer: C = 45°
04 Practice Questions
1A triangle has angle A=45°, angle B=45°, side a=10. Find side b.
b=10 (since angle A = angle B, sides are equal)
2A 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)
3What information do you need to apply the Law of Sines?
A matching angle-side pair, plus one more angle or side
4A triangle has angles 60° and 80°. Find the third angle.
40°
5What is the 'ambiguous case' in the Law of Sines?
When SSA information can produce zero, one, or two valid triangles
📄 Law of Sines — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.