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The sine and cosine laws for non-right triangles

By Daniel Whitman · Last updated April 2026 · 12 min read

The Pythagorean theorem solves any right-angled triangle. The two laws on this page solve the ones that are not right-angled. Together they cover essentially every triangle problem you can encounter at school, and the question is always the same: given some sides and angles, find the rest.

The convention for the rest of this article: a triangle has vertices $A$, $B$, $C$, and the side opposite each vertex is labelled with the matching lower-case letter. So $a$ is the side opposite vertex $A$.

The two laws

Law of sines

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

The ratio of a side to the sine of the angle opposite it is the same for all three pairs. The law of sines is the right tool when you know either two angles and any side (ASA / AAS) or two sides and a non-included angle (SSA — the awkward case discussed below).

Law of cosines

$$c^2 = a^2 + b^2 - 2ab\cos C.$$

This is the Pythagorean theorem with a correction term. When $C = 90^\circ$, $\cos C = 0$ and the correction disappears, leaving $c^2 = a^2 + b^2$ — the Pythagorean theorem. The law of cosines is the right tool when you know two sides and the included angle (SAS), or all three sides (SSS).

Choosing the right law: the four configurations

For each kind of information you start with, there is a clearly correct choice of law. Memorise this table once and the rest is mechanical.

The ambiguous case (SSA)

SSA is the configuration where things go wrong. Given sides $a$ and $b$ and angle $A$, the law of sines gives

$$\sin B = \frac{b \sin A}{a}.$$

This equation can have zero solutions, one solution, or two solutions for $B$ in $(0^\circ, 180^\circ)$, depending on the numbers:

The two-solution case is what gives SSA its nickname — the ambiguous case. Always check whether a second triangle is possible before reporting your answer.

The mistakes I see most often

1. Calculator in the wrong angle mode

Almost every wrong answer in trigonometry comes from a calculator set to radians when you wanted degrees, or vice versa. Check the mode indicator before you start each problem; on most calculators it is labelled DEG, RAD, or GRAD in the top corner of the display.

2. Forgetting the second SSA solution

If you compute $B = 30^\circ$ from $\sin B = 0.5$, the equation also has the solution $B = 150^\circ$. Whether the second one gives a valid triangle depends on whether $A + 150^\circ < 180^\circ$. Always check.

3. Using the wrong side opposite the angle

$\tfrac{a}{\sin A} = \tfrac{b}{\sin B}$ matches each side with the angle opposite it, not adjacent to it. Mismatching is the single most common arithmetic error in law-of-sines problems.

4. Sign of the cosine when the angle is obtuse

If you find $\cos C < 0$, then $C$ is obtuse (between $90^\circ$ and $180^\circ$). Some students reject the negative answer thinking they have made a sign error; they have not.

Where this comes up later

Vector arithmetic, force diagrams in mechanics, surveying, navigation, and any problem in three dimensions that has been reduced to a planar triangle. The cosine law in particular is the bridge between basic geometry and the dot product in linear algebra: $\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta$ is essentially the law of cosines rearranged.

References