The cosine rule is a formula used for calculating either an unknown side length or the size of an unknown angle.

The cosine rule

Maths

In the prior unit, pupils explored the trigonometric ratios for right-angled triangles in a variety of situations. In this unit, pupils explore what can be done when we have a triangle that is not right-angled.

Non right-angled trigonometry

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

Match the correct rules, theorem or formulae.

Which of the following are the cosine rule?

A triangle has adjacent lengths of 16 cm and 20 cm with an angle in between of 55°. Work out the missing length to 1 decimal place.

Given $$\cos(A)=\frac{r}{c}$$ and $$r=c\cos(A)$$

This proof has been mixed up. Write each line in order, starting with $$p^2 + (b - r)^2 = a^2$$.

Given a triangle with lengths labelled $$a$$, $$b$$ and $$c$$, match up the length $$b$$ with dimensions of the triangle and the angle $$B$$.

Given a triangle with lengths labelled $$a$$, $$b$$ and $$c$$, match up the angle $$B$$ with the dimensions of the triangle. 

$$\text{hyp}\times\cos(\theta°)=\text{adj}$$

$$\text{hyp}=\frac{\text{adj}}{\cos(\theta°)}$$

$$\cos(\theta°)=\frac{\text{hyp}}{\text{adj}}$$

Which of the following are equivalent to $$\cos(\theta°)=\frac{\text{adj}}{\text{hyp}}$$?

$$y$$-coordinate of P on the triangle formed inside the unit circle.

$$x$$-coordinate of P on the triangle formed inside the unit cirlce.

line that intersects the circle exactly once.

Match the trigonometric ratio with the correct definition.

A right-angled triangle has a height of 15 cm and a hypotenuse of 17 cm. Work out the area.

An isosceles triangle has a base of 16 cm and base angles of 53.97° (2 d.p). The area of this triangle, given to 2 significant figures, is {{}}cm$$^2$$.

Six equilateral triangles are put together to make a regular hexagon. The length of each triangle is 10 cm. Work out the exact area of the regular hexagon. 

This lesson will be removed by end of Summer Term 2025.

The cosine rule

Lesson outcome

This lesson will be removed by end of Summer Term 2025.