How to Use the Law of Sines: A Step-by-Step Guide

The Law of Sines is a powerful tool for solving any triangle when you know certain angles and sides. This guide walks you through manual calculations step by step, so you understand exactly how the formula works. If you prefer instant results, use our Law of Sines Calculator. For a deeper understanding of the formula itself, check out our Law of Sines Formula Explanation.

What You'll Need

• A scientific calculator (or one with sine function)
• Paper and pencil
• Knowledge of basic triangle angle sum (180°)
• Values for at least one side and its opposite angle, plus one more side or angle

Step-by-Step Method

Follow these steps to solve a triangle using the Law of Sines. The method works for both finding missing sides and missing angles.

1. Identify what you know. Determine which sides and angles are given. Label the triangle with vertices A, B, C and opposite sides a, b, c respectively. Note that the Law of Sines relates each side to the sine of its opposite angle.
2. Set up the proportion. Write the Law of Sines formula: a / sin(A) = b / sin(B) = c / sin(C). Choose the two ratios that involve the known values and the unknown you want to find.
3. If finding a side: Rearrange the proportion to isolate the unknown side. For example, if you know side a, angle A, and angle B, then side b = (a × sin(B)) / sin(A).
4. If finding an angle: Rearrange to solve for the sine of the unknown angle: sin(B) = (b × sin(A)) / a. Then take the inverse sine (sin⁻¹) to find the angle in degrees or radians.
5. Check for the ambiguous case. When given two sides and a non-included angle (SSA), there may be two possible triangles, one triangle, or none. We'll cover this more in the pitfalls section.
6. Find the third angle. Since all triangle angles sum to 180°, subtract the two known angles from 180° to get the third angle.
7. Find the remaining side(s). Use the Law of Sines again with the newly found angle to find any other unknown side.
8. Verify your results. Check that the sum of all sides and angles make sense. If you used the inverse sine, ensure the angle is acute (<90°) if required, or consider the ambiguous case.

Example 1: Finding a Missing Side (AAS)

Given: In triangle ABC, angle A = 40°, angle B = 60°, and side a = 10 cm. Find side b.

1. First, find angle C: C = 180° - 40° - 60° = 80°.
2. Use the Law of Sines: a / sin(A) = b / sin(B). So b = (a × sin(B)) / sin(A).
3. Plug in: sin(40°) ≈ 0.6428, sin(60°) ≈ 0.8660. b = (10 × 0.8660) / 0.6428 ≈ 13.47 cm.
4. So side b ≈ 13.5 cm. You could also find side c similarly.

Example 2: Finding a Missing Angle (SSA – Ambiguous Case)

Given: In triangle ABC, side a = 8 cm, side b = 12 cm, and angle A = 30°. Find angle B.

This is an SSA situation, so we must check for the ambiguous case.

1. Set up: sin(B) / b = sin(A) / a → sin(B) = (b × sin(A)) / a = (12 × 0.5) / 8 = 6/8 = 0.75.
2. Take inverse sine: B ≈ 48.6°.
3. But sin(180° - 48.6°) = sin(131.4°) also equals 0.75. So there is a second possible angle: B' = 180° - 48.6° = 131.4°.
4. Check if both possibilities form valid triangles. For B = 48.6°, then C = 180° - 30° - 48.6° = 101.4°, valid. For B = 131.4°, then C = 180° - 30° - 131.4° = 18.6°, also valid.
5. Thus there are two possible triangles. Use the Law of Sines again to find side c for each case.

See our Results Interpretation Guide for more on handling these cases.

Common Pitfalls

• Using degrees vs. radians: Make sure your calculator is in the correct mode. The Law of Sines works with any angle measure as long as you are consistent.
• Forgetting the ambiguous case (SSA): Always check if a second triangle is possible. Our calculator automatically detects this for you.
• Rounding errors: Keep as many decimals as possible during intermediate steps to avoid inaccuracies.
• Mismatching sides and angles: The side in the numerator must correspond to the angle inside the sine function. Double-check your labeling.

If you need a refresher on the basics, read our What is the Law of Sines? article. For real-world applications, see Law of Sines in Surveying & Navigation.