Overview

The Law of Sines is an important tool for solving non-right triangles. It connects side lengths and opposite angles in a clean ratio and is especially effective when at least one side-angle opposite pair is known.

For a triangle $\triangle ABC$ with side lengths $a,b,c$ opposite $A,B,C$ respectively, the law states:

where $R$ is the circumradius.

This topic appears often in olympiad geometry and trigonometric geometry because it lets you switch between angle information and length information quickly.

Key Ideas

• Opposite pairs matter: $a \leftrightarrow A$, $b \leftrightarrow B$, $c \leftrightarrow C$.
• The law is strongest when you know an angle and its opposite side.
• The circumradius form $a=2R\sin A$ is often the fastest path.
• Combine with $A+B+C=180^\circ$ to generate missing angles first.
• In SSA data (two sides and a non-included angle), check for the ambiguous case (0, 1, or 2 triangles).

Statement and Equivalent Forms

The standard ratio form is:

Equivalent pairwise forms are:

With circumradius $R$:

These forms are algebraically equivalent; choose the one that isolates the unknown most directly.

Why It Is True (Quick Derivation)

Use area formulas from different bases. Let $K$ be the area of $\triangle ABC$.

From $\frac12 bc\sin A=\frac12 ca\sin B$, cancel common factors:

Similarly,

So all are equal, giving the Law of Sines.

To get $\frac{a}{\sin A}=2R$, use the extended chord relation in a circle: a side of the triangle is a chord of the circumcircle.

When to Use Law of Sines vs. Law of Cosines

Use Law of Sines when:

• You know AAS/ASA data (two angles and a side).
• You know SSA data and want to test possible triangles.
• You need circumradius or expressions involving $2R$.

Use Law of Cosines when:

• You know SSS (all three sides).
• You know SAS and need the third side.
• You need an equation with $\cos$ of an included 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.

Solve for the Unknown with One Ratio

Do not use the full chain unless needed. Set up one proportion containing only one unknown and solve cleanly.

Use Angle Sum Early

If one angle is missing, compute it first using

Then apply Law of Sines with a known opposite pair.

Ambiguous Case (SSA)

Suppose you know $A$, $a$, and $b$, where $A$ is not included between known sides. From

you get

Now check possibilities:

• If $\frac{b\sin A}{a}\gt 1$, no triangle exists.
• If $\frac{b\sin A}{a}=1$, exactly one right-triangle case for $B$.
• If $0\lt\frac{b\sin A}{a}\lt 1$, then potentially two angles: $B_1=\arcsin\!\left(\frac{b\sin A}{a}\right)$ and $B_2=180^\circ-B_1$.

Keep only values where $A+B\lt 180^\circ$.

Worked Example

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

So

More Examples

Example 1: Find an Angle

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

Thus

Both may be valid depending on the remaining data. This is the SSA ambiguous case in action.

Example 2: Circumradius

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

so

Example 3: Find a Side After Angle Sum

Given $A=50^\circ$, $C=60^\circ$, and $c=14$, find $a$.

First,

Then apply Law of Sines with known pair $(c,C)$:

Strategy Checklist

• Label opposite sides and angles clearly before substituting.
• Use the triangle angle sum to compute a missing angle first.
• Keep unit of angle measurement consistent (degrees vs radians).
• Check whether Law of Cosines is a better fit for SSS/SAS.
• In SSA, explicitly test for 0/1/2 possible triangles.

Common Pitfalls

• Using degrees for some angles and radians for others.
• Swapping which side corresponds to which angle.
• Forgetting to test both $\theta$ and $180^\circ-\theta$ in SSA.
• Rounding too early and losing accuracy in later steps.

Summary

Law of Sines is a conversion tool between side lengths and opposite angles. Its most useful forms are

and

Use it by matching opposite pairs carefully, computing missing angles early, and checking the SSA case whenever applicable.

Practice Problems