Law of Sines — Formula & Solved Problems

The law of sines helps you solve a triangle when you know one side and its opposite angle. In any triangle with sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$,

$$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$$

The key rule is matching opposite pairs. Side $a$ goes with angle $A$, side $b$ with angle $B$, and side $c$ with angle $C$. If you mix up those pairs, the setup is wrong even if the algebra is fine.

What the law of sines means

The formula says every side-to-opposite-angle pair follows the same ratio. That is why a larger angle faces a longer side, while a smaller angle faces a shorter side.

That idea is the fastest intuition check. If one angle opens wider, the side across from it should be longer. If your answer breaks that pattern, you likely matched the wrong side and angle.

When to use the law of sines

The law of sines works for any triangle, but it is most useful for non-right triangles when you already know at least one opposite side-angle pair.

The most common setups are:

• AAS or ASA: two angles and one side
• SSA: two sides and a non-included angle, where the known angle is opposite one of the known sides

If you know two sides and the included angle instead, start with the law of cosines, not the law of sines.

Law of sines formula example

Suppose $A = 42^\circ$, $B = 71^\circ$, and $a = 8$. Find side $b$.

Start with matching opposite pairs:

$$\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$$

Substitute the known values:

$$\frac{8}{\sin(42^\circ)} = \frac{b}{\sin(71^\circ)}$$

Now solve for $b$:

$$b = 8 \cdot \frac{\sin(71^\circ)}{\sin(42^\circ)}$$

Using decimal approximations,

$$b \approx 8 \cdot \frac{0.9455}{0.6691} \approx 11.30$$

So

$$b \approx 11.3$$

This makes sense. Since $B$ is larger than $A$, side $b$ should be longer than side $a$, and $11.3 > 8$.

Common mistakes with the law of sines

The most common mistake is pairing a side with the wrong angle. The law of sines uses opposite pairs, not adjacent ones.

Another mistake is choosing it too early. If no opposite side-angle pair is known, it is usually not the best first equation.

Students also miss the SSA ambiguous case. If you get $\sin(B) = k$ with $0 < k < 1$, there can be two possible angles: $B$ and $180^\circ - B$.

That does not always mean two triangles exist. You must check whether each angle choice makes the full angle sum stay below $180^\circ$ and keeps the given side data consistent.

Two equivalent law of sines forms

You may see the law of sines written in either of these forms:

$$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} \quad \text{or} \quad \frac{\sin(A)}{a} = \frac{\sin(B)}{b}$$

They mean the same thing. Choose the version that isolates the unknown most cleanly, but keep the opposite-pair matching rule the same.

Where the law of sines is used

The law of sines appears in trigonometry, geometry, surveying, navigation, and any triangle measurement problem where no right angle is given.

In practice, the workflow is simple: draw the triangle, label opposite pairs, check whether the known information fits ASA, AAS, or SSA, and then solve.

Try a similar problem

Try your own version with $A = 35^\circ$, $C = 95^\circ$, and $a = 12$. First find angle $B$, then use the law of sines to find side $c$. Before calculating, predict whether $c$ should be longer or shorter than $a$. That quick prediction is one of the easiest ways to catch a setup mistake early.