Mathematics

Sine & cosine rules

Even oblique triangles without a right angle can be fully calculated.

What you need first

, and tangent need a right angle. But most triangles in the world have none. Two rules close this gap: the sine rule and the cosine rule hold in every triangle and let you find missing sides and angles, no matter how oblique it is.

Naming sides and angles

In any triangle a side and an angle face each other: side $a$ lies opposite angle $\alpha$ , $b$ opposite $\beta$ , $c$ opposite $\gamma$ . This pairing is the key to both rules, because they always link a side with the angle facing it.

Triangle lab

Leg a3

Leg b4

3² + 4² = 25 → c = √25 = 5

Try it: this is the special case with a right angle, where c² = a² + b² holds. The cosine rule will turn it into a formula that works for every triangle shape.

The sine rule

The sine rule says every side stands in the same ratio to the sine of its opposite angle. It helps when you know one side with its opposite angle plus one more side or angle. Example: $b = 12$ , $\beta = 35°$ and $\alpha = 70°$ give $a = \frac{b \cdot \sin\alpha}{\sin\beta} \approx 19.66$ .

\frac{a}{\sin\alpha} = \frac{b}{\sin\beta} = \frac{c}{\sin\gamma}

Sine rule: each side divided by the sine of its opposite angle gives the same value.

The cosine rule

If you know two sides and the angle between them, the cosine rule helps. It extends the Pythagorean theorem: at a right angle $\cos\gamma = 0$ and the last term drops away, leaving $c^2 = a^2 + b^2$ . Example: $a = 9$ , $b = 5$ and $\gamma = 60°$ give $c^2 = 81 + 25 - 90 \cdot 0.5 = 61$ , so $c \approx 7.81$ . If the angle is obtuse, that is larger than 90°, its cosine turns negative and the last term is added instead of subtracted. The third side then comes out longer.

c^2 = a^2 + b^2 - 2ab\cos\gamma

Cosine rule: the third side from two sides and the included angle.

Looking for angles instead of sides

Both rules can be rearranged for an angle as well. If you know all three sides, rearrange the cosine rule for $\cos\gamma$ , and the inverse function $\cos^{-1}$ turns that value into the angle. If the cosine comes out negative, the angle is obtuse. If instead you know a side with its opposite angle plus a second side, the sine rule gives $\sin\alpha = \frac{a \cdot \sin\beta}{b}$ , and $\sin^{-1}$ turns that into the angle you want. As always with angles: your calculator has to be set to degrees (DEG).

\cos\gamma = \frac{a^2 + b^2 - c^2}{2ab}

The cosine rule rearranged for the angle: this turns three sides into the angle γ.

Exercises

0 of 6 solved

Time to try it yourself. You can't break anything, every attempt counts.

Which rule helps when you know two sides and the included angle?

What is the sine rule?

In a triangle b = 10, β = 40° and α = 60°. How long is side a? Round to two decimal places.

Match each side to the angle that lies opposite it in the triangle.

Side a

Side b

Side c

In a triangle a = 5, b = 7 and the included angle γ = 45°. How long is side c? Round to two decimal places.

At a right angle cos γ becomes …, so the last term drops away and the cosine rule turns into the Pythagorean theorem.

Report an error