# Section 4: Sine And Cosine Rule

## Introduction

This section will cover how to:
• Use the Sine Rule to find unknown sides and angles
• Use the Cosine Rule to find unknown sides and angles
• Combine trigonometry skills to solve problems

Each topic is introduced with a theory section including examples and then some practice questions. At the end of the page there is an exercise where you can test your understanding of all the topics covered in this page.

You are allowed to use calculators in this topic.
All answers should be given to 3 significant figures unless otherwise stated.

## Formulae You Should Know

You should already know each of the following formulae:
 formulae for right-angled triangles formulae for all triangles

NOTE: The only formula above which is in the A Level Maths formula book is the one highlighted in yellow.
You must learn these formulae, and then try to complete this page without referring to the table above.

## Sine Rule

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 top:
 a = b sin(A) sin(B)
You will only ever 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 1Start by writing out the Sine Rule formula for finding sides:
 a = b sin(A) sin(B)
Step 2Fill in the values you know, and the unknown length:
 x = 7 sin(80°) sin(60°)
Remember that each fraction in the Sine Rule formula should contain a side and its opposite angle.
Step 3Solve the resulting equation to find the unknown side, giving your answer to 3 significant figures:
 x = 7 (multiply by sin(80°) on both sides) sin(80°) sin(60°) x = 7 × sin(80°) sin(60°) x = 7.96 (accurate to 3 significant figures)
Note that 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 Sine Rule where the angles are on the top:
 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 1Start by writing out the Sine Rule formula for finding angles:
 sin(A) = sin(B) a b
Step 2Fill in the values you know, and the unknown angle:
 sin(m°) = sin(75°) 8 10
Remember that each fraction in the Sine Rule formula should contain a side and its opposite angle.
Step 3Solve the resulting equation to find the sine of the unknown angle:
 sin(m°) = sin(75°) (multiply by 8 on both sides) 8 10 sin(m°) = sin(75°) × 8 10 sin(m°) = 0.773 (3 significant figures)
Step 4Use the inverse-sine function (sin–1) to find the angle:
 m° = sin–1(0.773) = 50.6° (3sf)
Other Notes
 You may be aware that sometimes Sine Rule questions can have two solutions (only when you are finding angles) – you do not need to know about these additional solutions at this time but you will learn more about them next year.
Practice Questions
 Work out the answer to each question then click on the button marked to see if you are correct.

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

 a = b sin(A) sin(B) p = 21 × sin(32°) both sides sin(32°) sin(95°) p = 21 × sin(32°) sin(95°) p = 11.2 (accurate to 3 significant figures)

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

 sin(A) = sin(B) a b sin(b°) = sin(100°) × 3.6 both sides 3.6 5.1 sin(b°) = sin(100°) × 3.6 5.1 sin(b°) = 0.695 (3sf) b° = sin-1(0.695) = 44.0° (3sf)

## Cosine Rule

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 a2 is the subject of the formula:
 a2 = b2 + c2 – 2bc 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 1Start by writing out the Cosine Rule formula for finding sides:
 a2 = b2 + c2 – 2bc cos(A)
Step 2Fill in the values you know, and the unknown length:
 x2 = 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 3Evaluate the right-hand-side and then square-root to find the length:
 x2 = 222 + 282 – 2×22×28×cos(97°) (evaluate the right hand side) x2 = 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) = b2 + c2 – a2 2bc
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 1Start by writing out the Cosine Rule formula for finding angles:
 cos(A) = b2 + c2 – a2 2bc
Step 2Fill 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 3Evaluate 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)
Other Notes
 If you know two sides and an angle which is not inbetween them then you can use the Cosine Rule to find the other side, but it is easier to use the Sine Rule in this situation – you should always use the Sine Rule if you have an angle and its opposite side.
Practice Questions
 Work out the answer to each question then click on the button marked to see if you are correct.

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

 a2 = b2 + c2 – 2bc cos(A) h2 = 882 + 1462 – 2×88×146×cos(53°) h2 = 13595.761..... h2 = 117 (accurate to 3 significant figures)

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

 cos(A) = b2 + c2 – a2 2bc cos(a°) = 3.12 + 4.32 – 5.92 2 × 3.1 × 4.3 cos(a°) = – 0.252 a° = cos–1(– 0.252) a° = 105° (3sf)

## Combining Trigonometry Skills

Choosing The Appropriate Technique
 Sometimes more than one technique from the formula table at the top of this page can be used to solve a trig problem, but you will want to choose the most efficient and easiest method to save time. The flowchart below shows how to decide which method to use:
Examples

These examples illustrate the decision-making process for a variety of triangles:

 e.g. 1 The triangle is not right-angled. We do know a side and its opposite angle. Therefore we use the Sine Rule. e.g. 2 The triangle is right-angled. The question involves angles. Therefore we use trig ratios - sin, cos and tan. e.g. 3 The triangle is right-angled. The question does not involve angles. Therefore we use Pythagoras's Theorem. e.g. 4 The triangle is not right-angled. We do not know a side and its opposite angle. Therefore we use the Cosine Rule.
Practice Questions
 Work out the answer to each question then click on the button marked to see if you are correct.
Find the unknown side or angle in each of the following diagrams:

(a)

The triangle is not right-angled, and we don't know a side and its opposite angle, so we need to use the Cosine Rule.
 cos(a°) = 42 + 72 – 62 2 × 4 × 7 cos(a°) = 0.518 a° = cos–1(0.518) a° = 58.8° (3sf)

(b)

The triangle is right-angled, and the question involves angles, so we need to use trigonometric ratios.
 cos(50°) = y 8
 8 × cos(50°) = y y = 5.14 (3sf)

(c)

The triangle is not right-angled, but we do know a side and its opposite angle, so we use the Sine Rule.
 sin(b°) = sin(62°) 11 10 sin(b°) = sin(62°) × 11 10 sin(b°) = 0.971 (3sf) b° = sin-1(0.971) = 76.2° (3sf)

(d)

The triangle is right-angled, but the question does not involve angles, so we need to use Pythagoras's Theorem.
 x2 = 7.52 + 182 x2 = 380.25 x = 19.5

## Exercise

Work out the answers to the questions below and fill in the boxes. Click on the
button to find out whether you have answered correctly. If you have then the answer will be ticked
and you should move on to the next question. If a cross
to clear the incorrect answer and have another go, or you can click on
to get some advice on how to work out the answer and then have another go. If you still can't work out the answer after this then you can click on
to see the solution.

#### Questions

Work out the values of x, y and z for each of the diagrams below. The diagrams are not to scale.
Try and use the most efficient method you can and remember to give your answer to 3 significant figures.

(a)

x

=
° The angles in a triangle add up to 180° 180° – 34° – 115° = 31°

y

=
Use the Sine Rule
 y = 13 sin(31°) sin(115°) y = 13 × sin(31°) sin(115°) y = 7.39

z

=
Use the Sine Rule again
 z = 13 sin(34°) sin(115°) z = 13 × sin(34°) sin(115°) z = 8.02

(b)

x

=
° Use the Cosine Rule
 cos(x°) = 62 + 102 – 122 2 × 6 × 10 cos(x°) = –0.0667 x° = cos–1(–0.0667) x° = 93.8°

y

=
° Use the Cosine Rule again
 cos(y°) = 62 + 122 – 102 2 × 6 × 12 cos(y°) = 0.556 y° = cos–1(0.556) y° = 56.3°

z

=
° The angles in a triangle add up to 180° 180° – 93.8° – 56.3° = 29.9°

(c)

x

=
° The angles in a triangle add up to 180° 180° – 75° – 65° = 40°

y

=
Use the Sine Rule
 y = 9 sin(75°) sin(65°) y = 9 × sin(75°) sin(65°) y = 9.59

z

=
Use the Sine Rule again
 z = 9 sin(40°) sin(65°) z = 9 × sin(40°) sin(65°) z = 6.38

(d)

x

=
The triangle to the right of the dotted line is right-angled, so use the appropriate trig ratio (sin, cos or tan)
 x = sin(70°) 6 x = sin(70°) × 6 x = 5.64

y

=
° The triangle to the left of the dotted line is also right-angled, so use the appropriate trig ratio (sin, cos or tan)
 cos(y°) = 5.64 8 cos(y°) = 0.705 y° = cos–1(0.705) y° = 45.2°

z

=
° The angles in a triangle add up to 180° 180° – 90° – 45.2° = 44.8°

(e)

x

=
° Use the Sine Rule
 sin(x°) = sin(54°) 5 7 sin(x°) = sin(54°) × 5 7 sin(x°) = 0.578 x° = sin-1(0.578) = 35.3°

y

=
° The angles in a triangle add up to 180° 180° – 54° – 35.3° = 90.7°

z

=
Use the Sine Rule again
 z = 7 sin(90.7°) sin(54°) z = 7 × sin(90.7°) sin(54°) z = 8.65

(f)

x

=
° Use the Cosine Rule
 cos(x°) = 62 + 62 – 92 2 × 6 × 6 cos(x°) = –0.125 x° = cos–1(–0.125) x° = 97.2°

y

=
° Don't use the Cosine Rule again – this is an isosceles triangle (180° – 97.2°) ÷ 2 = 41.4°

z

=
° The angles in a triangle add up to 180° 180° – 97.2° – 41.4° = 41.4°

(g)

x

=
° Neither right-angled triangle either side of the dotted line has enough information, so you will need to use Cosine Rule on the outside triangle
 cos(x°) = 122 + 202 – 152 2 × 12 × 20 cos(x°) = 0.665 x° = cos–1(0.665) x° = 48.3°

y

=
After finding x you have enough information to use trigonometry in the right-angled triangle above the dotted line
 y = sin(48.3°) 12 y = sin(48.3°) × 12 y = 8.97

z

=
° After finding y you have enough information to use trigonometry in the right-angled triangle below the dotted line
 sin(z°) = 8.97 15 sin(z°) = 0.598 z° = sin–1(0.598) z° = 36.7°

(h)

( definitely not to scale! )

x

=
° Use the Sine Rule
 sin(x°) = sin(54°) 4 9 sin(x°) = sin(54°) × 4 9 sin(x°) = 0.360 x° = sin-1(0.360) = 21.1°

y

=
° The angles in a triangle add up to 180° 180° – 54° – 21.1° = 105° (3sf)

z

=
Use the Sine Rule again
 z = 9 sin(104.9°) sin(54°) z = 9 × sin(104.9°) sin(54°) z = 10.7