10.07 The cosine rule

If the three side lengths in a triangle are a, b and c, with an angle C opposite the side with length c, then c^2=a^2+b^2-2ab\cos C.

The cosine rule relates the lengths of all three sides in a triangle and the cosine of one of its angles. Therefore, the cosine rule will help us to find:

• the third side of a triangle when you know two sides and the included angle (the angle between the two known sides)

• the angles of a triangle when you know all three sides

This rule can be proven using Pythagoras' theorem and right-angled trigonometry.

What happens when the angle C is 90\degree? This means the triangle is right-angled, and the side c is the hypotenuse. Putting this into the cosine rule, we get:

which is Pythagoras' theorem. This is why we say that the cosine rule is a generalisation of Pythagoras' theorem.

Note: The formula can be written in terms of any of the sides or angles, familiarise yourself with the forms below and note the pattern. If a triangle is labelled differently we can adapt the rule using the pattern.

The actual letters used in the formula do not have to be the same as the triangle. The convention is that the lengths of sides are given in lowercase letters a,b,c,q,u,v,s,t and the size of the angles given in capital letters A,B,C,Q,U,V,S,T. \begin{aligned} c^2&=a^2+b^2-2ab\cos C\\ v^2&=s^2+t^2-2st\cos V\\ w^2&=u^2+v^2-2uv\cos W \end{aligned}

Examples

Example 1

Find the length of a using the cosine rule. Round your answer to two decimal places.

Worked Solution

Create a strategy

Use the cosine rule a^2=b^2+c^2-2bc\cos A.

Apply the idea

\displaystyle a^2	\displaystyle =	\displaystyle b^2+c^2-2bc\cos A	Write the formula
\displaystyle a^2	\displaystyle =	\displaystyle 13^2+30^2-2\times 13\times 30\cos 33\degree	Substitute \, b=13, \, c=30, \, A=33
	\displaystyle \approx	\displaystyle 414.83696	Evaluate
\displaystyle a	\displaystyle \approx	\displaystyle \sqrt{414.83696}	Take the square root of both sides
\displaystyle a	\displaystyle \approx	\displaystyle 20.37	Evaluate and round

Watch question walkthrough

To find a missing angle using the cosine rule, we need to know all three side lengths.

When looking to solve for an unknown angle, the equation can also be rearranged and written as \cos C=\dfrac{a^2+b^2-c^2}{2ab}.

Examples

Example 2

Find the value of angle w using the cosine rule. Write your answer correct to two decimal places.

Worked Solution

Create a strategy

Use the rearranged formula of the cosine rule given by \cos C=\dfrac{a^2+b^2-c^2}{2ab}, where w corresponds to C.

Apply the idea

\displaystyle \cos C	\displaystyle =	\displaystyle \dfrac{a^2+b^2-c^2}{2ab}	Write the formula
\displaystyle \cos w	\displaystyle =	\displaystyle \frac{32^2+19^2-25^2}{2\times 32\times 19}	Susbtitute the values
\displaystyle \cos w	\displaystyle =	\displaystyle \dfrac{5}{8}	Evaluate
\displaystyle w	\displaystyle =	\displaystyle \cos ^{-1}\left(\frac{5}{8}\right)	Take the inverse cosine of both sides
	\displaystyle \approx	\displaystyle 51.32\degree	Evaluate and round

Watch question walkthrough