How Do You Use the Cosine Rule for a Non-Right Triangle?

Use the cosine rule when a non-right triangle gives you either two sides and the included angle (SAS) or all three sides (SSS). It allows you to calculate a missing side or angle without needing a right angle.

Choosing the Correct Form

Label each side opposite its corresponding angle: side $a$ is opposite angle $A$, and similarly for $b$, $B$, $c$, and $C$.

The same structure applies whichever side or angle is unknown. For example, to find $a$, use $a^2=b^2+c^2-2bc\cos A$.

The rule extends Pythagoras' theorem by including the correction term $-2ab\cos C$. When $C=90^\circ$, $\cos C=0$, so it becomes .

Worked Example

Suppose $a=7$, $b=10$, and the included angle is $C=60^\circ$. To find the opposite side $c$, substitute into the cosine rule:

$$c^2=7^2+10^2-2(7)(10)\cos 60^\circ$$ $$c^2=49+100-140\left(\frac{1}{2}\right)=79$$

Therefore,

$$c=\sqrt{79}\approx 8.89.$$

The positive square root is used because a side length must be positive. If finding an angle instead, first calculate its cosine value and then apply inverse cosine, such as $C=\arccos(\cdots)$.

For an SSS example, if $a=5$, $b=7$, and $c=8$, then $\cos C=\frac{5^2+7^2-8^2}{2(5)(7)}=$, giving . This angle is opposite side , the longest side, so it should be the largest angle. This provides a useful reasonableness check before finalizing the answer.

A common misconception is that any two known sides and any angle are sufficient. The angle must be the included angle between the two known sides when finding the third side; otherwise, the sine rule may be more appropriate.

Exam Technique

This is IB Mathematics AA SL 3.2 and may be tested on Paper 1 or Paper 2. Show the substituted cosine-rule equation before calculating, check that each angle is paired with its opposite side, and give an exact answer or round to three significant figures unless instructed otherwise.