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NOTE: The only formula above which is in the A Level Maths formula book is the one highlighted in yellow.

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The Sine Rule can be used in any triangle (not just right-angled triangles) where a side and its opposite angle are known.

Finding Sides

If you need to find the length of a side, you need to use the version of the Sine Rule where the lengths are on the vertex:

	a	=	b
sin(A)	sin(B)

You will only need two parts of the Sine Rule formula, not all three. You will need to know at least one pair of a side with its opposite angle to use the Sine Rule.

Finding Sides Example

Work out the length of x in the diagram below:

Step 1

Start by writing out the Sine Rule formula for finding sides:

	a	=	b
sin(A)	sin(B)

Step 2

Fill in the values you know, and the unknown length:

	x	=
sin(80°)	sin(60°)

Remember that each fraction in the Sine Rule formula should contain a side and its opposite angle.

Step 3

Solve the resulting equation to find the unknown side, giving your answer to 3 significant figures:

	x	=		(multiply by sin(80°) on both sides)
sin(80°)	sin(60°)
	x	=		× sin(80°)
sin(60°)
	x	=	7.96 (accurate to 3 significant figures)

Finding Angles

If you need to find the size of an angle, you need to use the version of the Sine Rule where the angles are on the vertex:

	sin(A)	=	sin(B)
a	b

As before, you will only need two parts of the Sine Rule, and you still need at least a side and its opposite angle.

Finding Angles Example

Work out angle m ° in the diagram below:

Step 1

Start by writing out the Sine Rule formula for finding angles:

	sin(A)	=	sin(B)
a	b

Step 2

Fill in the values you know, and the unknown angle:

	sin(m °)	=	sin(75°)

Remember that each fraction in the Sine Rule formula should contain a side and its opposite angle.

Step 3

Solve the resulting equation to find the sine of the unknown angle:

sin(m °)	=	sin(75°)	(multiply by 8 on both sides)

sin(m °)	=	sin(75°)	× 8

sin(m °)	=	0.773 (3 significant figures)

Step4

Use the inverse-sine function (sin^–1) to find the angle:

m °

sin^–1(0.773) = 50.6° (3sf)

Practice Questions

(a) Find the missing side in the diagram below:

(b) Find the missing angle in the diagram below:

The Cosine Rule can be used in any triangle where you are trying to relate all three sides to one angle.

Finding Sides

If you need to find the length of a side, you need to know the other two sides and the opposite angle. You need to use the version of the Cosine Rule where a ² is the subject of the formula:

a ² = b ² + c ² – 2 bc cos(A)

Side a is the one you are trying to find. Sides b and c are the other two sides, and angle A is the angle opposite side a.

Finding Sides Example

Work out the length of x in the diagram below:

Step 1

Start by writing out the Cosine Rule formula for finding sides:

a ² = b ² + c ² – 2 bc cos(A)

Step 2

Fill in the values you know, and the unknown length:

x ² = 22² + 28² – 2× 22× 28× cos(97°)

It doesn't matter which way around you put sides b and c – it will work both ways.

Step 3

Evaluate the right-hand-side and then square-root to find the length:

x ²	=	22² + 28² – 2× 22× 28× cos(97°)	(evaluate the right hand side)
x ²	=	1418.143.....	(square-root both sides)
x	=	37.7 (accurate to 3 significant figures)

As with the Sine Rule you should try and keep full accuracy until the end of your calculation to avoid errors.

Finding Angles

If you need to find the size of an angle, you need to use the version of the Cosine Rule where the cos(A) is on the left:

cos(A)	=	b ² + c ² – a ²
2 bc

It is very important to get the terms on the top in the correct order; b and c are either side of angle A which you are trying to find and these can be either way around, but side a must be the side opposite angle A.

Finding Angles Example

Work out angle P ° in the diagram below:

Step 1

Start by writing out the Cosine Rule formula for finding angles:

cos(A)	=	b ² + c ² – a ²
2 bc

Step 2

Fill in the values you know, and the unknown length:

cos(P °)	=	5² + 8² – 7²
2 × 5 × 8

Remember to make sure that the terms on top of the fraction are in the correct order.

Step 3

Evaluate the right-hand-side and then use inverse-cosine (cos^–1) to find the angle:

cos(P °)	=	5² + 8² – 7²	(evaluate the right-hand side)
2 × 5 × 8
cos(P °)	=	0.5	(do the inverse-cosine of both sides)
P °	=	cos^–1(0.5) = 60° (3sf)

Practice Questions

(a) Find the missing side in the diagram below:

(b) Find the missing angle in the diagram below:

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