Overview

The Law of Sines is ideal for triangle problems with angles and one side. It also connects directly to circumradius.

Key Ideas

• $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$.
• Use it when you know a side-angle pair.
• It pairs well with the triangle angle sum to find a missing angle.

Core Skills

Match Each Side to Its Opposite Angle

Label the triangle so that $a$ is opposite $A$, and so on, before writing the ratio. This avoids the most common mistake.

Use the Circumradius Form

If $R$ is known or requested, use $a=2R\sin A$ directly.

Decide When It Is Appropriate

This law is strongest when you have an angle and its opposite side, or two angles and a side. If all three sides are known, the Law of Cosines is cleaner.

Worked Example

In $\triangle ABC$, $A=30^\circ$, $B=45^\circ$, and $a=10$. Find $b$.

$\frac{10}{\sin 30^\circ} = \frac{b}{\sin 45^\circ}$ gives $20 = b\sqrt{2}$, so $b=10\sqrt{2}$.

More Examples

Example 1: Find an Angle

In $\triangle ABC$, $a=8$, $b=10$, and $A=30^\circ$. Find $B$.

$\frac{8}{\sin 30^\circ} = \frac{10}{\sin B}$ gives $16 = 10/\sin B$, so $\sin B = 5/8$.

Example 2: Circumradius

If $a=12$ and $A=45^\circ$, find $R$.

$12 = 2R\sin 45^\circ$ gives $R = 12/(\sqrt{2}) = 6\sqrt{2}$.

Strategy Checklist

• Label opposite sides and angles clearly before substituting.
• Use the triangle angle sum to compute a missing angle first.
• Keep calculator mode consistent (degrees vs radians).
• Check whether the Law of Cosines is a better fit.

Common Pitfalls

• Using degrees for some angles and radians for others.
• Swapping which side corresponds to which angle.
• Forgetting that two angles might lead to an ambiguous case if not careful.

Practice Problems