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

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

Considering the appropriate trigonometric 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

Which of these formulae are the correct rearrangements where the subject of the formula is a length?

$$A=\cos^{-1}\left(\frac{b^2+c^2-a^2}{2bc}\right)$$

$$A=\sin^{-1}\left(\frac{a\times\sin(B)}{b}\right)$$

Which of these formulae are the correct rearrangements where the subject of the formula is angle $$A$$?

When you have two pairs of opposites, where one value in a pair is unknown

When three lengths of the triangle are given

When two adjacent lengths and the angle in between are given

Given a triangle with vertices labelled $$A$$, $$B$$ and $$C$$, how do you know you are able to use the sine rule?

Given a triangle with vertices labelled $$A$$, $$B$$ and $$C$$, how do you know you are able to use the cosine rule?

A triangle has 3 vertices, $$A$$, $$B$$ and $$C$$. It also has the corresponding lengths $$a$$, $$b$$ and $$c$$. When $$a$$ = 8.25 cm, $$C$$ = 76.0°, $$b$$ = 8 cm, work out the perimeter.

A triangle has vertices, $$N$$, $$L$$ and $$M$$. It has corresponding lengths $$n$$, $$l$$ and $$m$$. When $$M = 94.4°, n = 8.9$$ cm, $$N = 53.8°$$ and $$l = 5.8$$ cm, work out the perimeter (1 d.p.).

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

Match the correct rules, theorem or formulae.

Given a triangle with lengths labelled $$a=8$$ cm, $$B=72$$° and $$c=12$$ cm work out length $$b$$ cm.

Given a triangle with lengths labelled $$a=32$$ cm, $$b=24$$ cm and $$c=40$$ cm, work out angle $$B$$.

A plane starts at City X and flies 4 km due north to City A, then turns at a bearing of 068 for 10.8 km to City B. If the plane flew directly from City X to City B, what is the distance?

A plane starts at an airport and then travels North 8 km, then North East 4.2 km. What is the distance the plane needs to travel to get back to the airport?

A plane starts at an airport and then travels North 8 km, then North East 4.2 km. What is the bearing the plane needs to travel at to get back to the airport?

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

Considering the appropriate trigonometric rule

Lesson outcome

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