Law of Sines — Formula & Solved Problems

When a triangle gives you one side together with the angle directly across from it, the law of sines is usually the fastest way in. It connects every side to the angle facing it, so a single known pair unlocks the rest.

The formula and its symbols

In any triangle with sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$,

The symbols travel in opposite pairs: side $a$ with angle $A$, side $b$ with angle $B$, side $c$ with angle $C$. Matching those pairs correctly is the whole game. If you pair a side with the wrong angle, the setup is broken even when the algebra that follows is flawless.

You may also see the relationship flipped:

Both say the same thing. Pick whichever version isolates your unknown most cleanly, and keep the opposite-pair rule unchanged.

Why the ratio holds

The formula says every side-to-opposite-angle pair shares one common ratio. That is why a wider angle always faces a longer side, and a narrower angle faces a shorter side.

This gives you a built-in sanity check. If one angle opens wider, the side across from it should come out longer. When an answer breaks that pattern, you have almost certainly matched the wrong side and angle.

Worked example

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

Match opposite pairs:

Substitute the known values:

Solve for $b$:

Using decimal approximations,

So

This passes the check. Since $B$ is larger than $A$, side $b$ should exceed side $a$, and $11.3 > 8$.

Practice it yourself

Set up your own triangle with $A = 35^\circ$, $C = 95^\circ$, and $a = 12$. First find angle $B$ from the fact that the three angles sum to $180^\circ$, then apply the law of sines to find side $c$. Before you compute, predict whether $c$ should be longer or shorter than $a$ — that quick prediction catches setup slips before they cost you.

When you finish, confirm that the longest side faces the largest angle. If $C = 95^\circ$ is the widest angle, side $c$ should be the longest side in the triangle.

Where it applies and what trips students up

The law of sines works for any triangle, but it shines on non-right triangles where at least one opposite side-angle pair is already known. The common entry points 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 instead you know two sides and the included angle, start with the law of cosines, not the law of sines.

The pitfalls cluster around three slips. The first is pairing a side with the wrong angle — the law uses opposite pairs, never adjacent ones. The second is reaching for it too early; if no opposite pair is known, it is rarely the right first equation. The third is the SSA ambiguous case: if you get $\sin(B) = k$ with $0 < k < 1$, two angles may satisfy it, $B$ and $180^\circ - B$. That does not guarantee two triangles. You still must check that each choice keeps the angle sum below $180^\circ$ and stays consistent with the given side data.