The Sine Rule

What you'll learn

When the sine rule is useful for non-right-angled triangles.
How to match each side with its opposite angle.
How to find missing sides and missing angles using the sine rule.
How to combine the sine rule with perimeter and area questions.

Before you start: the triangle facts you need

The sine rule is used in triangles that are not necessarily right-angled. You do not need a 90° angle, and you usually cannot use SOHCAHTOA directly.

The most important skill is spotting opposite pairs: an angle and the side directly across from it.

A labelled triangle showing angles A, B, C and opposite sides a, b, c

Definition

Opposite side

In a triangle, the opposite side to an angle is the side across the triangle from that angle. In triangle ABC, side $a$ is opposite angle $A$ , side $b$ is opposite angle $B$ , and side $c$ is opposite angle $C$ .

You will also use the angle sum of a triangle:

180^\circ

Example

Finding the missing angle and an opposite pair

A triangle has angles 37° and 98°. A side of length 9 cm is opposite the 98° angle. Find the third angle and identify the side-angle pair.

Use the fact that angles in a triangle add to 180°:
$37∘+98∘=135∘37^\circ + 98^\circ = 135^\circ$
Subtract from 180°:
$180∘−135∘=45∘180^\circ - 135^\circ = 45^\circ$
The third angle is 45°.
The side of length 9 cm is paired with the 98° angle, because they are opposite each other.

Common Mistake

Do not pair an angle with a touching side

The sine rule only works with opposite pairs. If an angle touches a side, that does not mean they are paired for the sine rule.

The sine rule

The sine rule links the sides and angles of any triangle.

Definition

The sine rule

For triangle ABC, where sides $a$ , $b$ , and $c$ are opposite angles $A$ , $B$ , and $C$ :

asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

You can also write it as:

sin⁡Aa=sin⁡Bb=sin⁡Cc\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}

Key Idea

The big idea

Use the sine rule when you know one complete opposite pair, and you want another side or angle from a second opposite pair.

A complete opposite pair means you know both the side and the angle opposite it.

Finding a missing side

When finding a side, it is usually easiest to put the side lengths on top:

angle)\frac{\text{side}}{\sin(\text{opposite angle})}

You need:

one known side with its opposite angle;
the angle opposite the side you want.

Example

Finding a missing length

In a triangle, a side of length 12 cm is opposite an angle of 44°. Another side, $y$ cm, is opposite an angle of 101°. Find $y$ to 1 decimal place.

Match the opposite pairs:
- 12 cm is opposite 44°.
- $y$ cm is opposite 101°.
Set up the sine rule with sides on top:
$ysin⁡101∘=12sin⁡44∘\frac{y}{\sin 101^\circ} = \frac{12}{\sin 44^\circ}$
Multiply both sides by $sin⁡101∘\sin 101^\circ$ :
$\frac{12 \sin 101^\circ}{\sin 44^\circ}$
Calculate:
$16.96\ldots$
Round to 1 decimal place: $y = 17.0$ cm.

Tip

Calculator mode

Make sure your calculator is in degrees mode, not radians. In degrees mode, $sin⁡44∘\sin 44^\circ$ means the sine of 44°.

Finding a missing angle

When finding an angle, it is usually easier to put the sine parts on top:

sin⁡Aa=sin⁡Bb\frac{\sin A}{a} = \frac{\sin B}{b}

After finding $sin⁡A\sin A$ , use the inverse sine button on your calculator. This is often written as $sin^{-1}$ .

Definition

Inverse sine

The inverse sine, $sin^{-1}$ , finds the angle when you already know its sine value.

Example

Finding a missing angle

In a triangle, a side of length 8.1 cm is opposite angle $x$ . A side of length 12.6 cm is opposite an angle of 72°. Find $x$ to 3 significant figures.

Match the opposite pairs:
- 8.1 cm is opposite angle $x$ .
- 12.6 cm is opposite 72°.
Use the sine rule with sine parts on top:
$sin⁡x8.1=sin⁡72∘12.6\frac{\sin x}{8.1} = \frac{\sin 72^\circ}{12.6}$
Multiply both sides by 8.1:
$sin⁡x=8.1sin⁡72∘12.6\sin x = \frac{8.1 \sin 72^\circ}{12.6}$
Calculate the sine value:
$sin⁡x=0.611…\sin x = 0.611\ldots$
Use inverse sine:
$\sin^{-1}(0.611\ldots)$
Round to 3 significant figures: $37.7^\circ$ .

Common Mistake

Rounding too early

Do not round the sine value halfway through the question. Keep the full calculator value until the final answer, then round as requested.

Sometimes you must find a missing angle first

Some questions give two angles and one side. If the angle opposite the known side is not directly given, find the third angle first.

Example

Using the angle sum before the sine rule

In triangle ABC, angle $A$ is 36°, angle $B$ is 82°, and side AB is 7 m. Find side AC to 3 significant figures.

Work out angle $C$ because side AB is opposite angle $C$ :
$180^\circ - 36^\circ - 82^\circ = 62^\circ$
Match the pairs:
- AB = 7 m is opposite angle $62^\circ$ .
- AC is opposite angle $82^\circ$ .
Use the sine rule:
$ACsin⁡82∘=7sin⁡62∘\frac{AC}{\sin 82^\circ} = \frac{7}{\sin 62^\circ}$
Rearrange and calculate:
$\frac{7 \sin 82^\circ}{\sin 62^\circ} = 7.85\ldots$
To 3 significant figures, AC = 7.85 m.

Perimeter questions

A perimeter is the total distance around the outside of a shape. For a triangle, add the three side lengths.

In sine rule perimeter questions, you may need to find two missing sides first.

Example

Finding the perimeter of a triangle

Triangle ABC has AC = 14.2 m, angle $B$ is 102°, and angle $C$ is 44°. Find the perimeter to 3 significant figures.

First find angle $A$ :
$180^\circ - 102^\circ - 44^\circ = 34^\circ$
Match the known pair:
- AC = 14.2 m is opposite angle $102^\circ$ .
Find BC, which is opposite angle $34^\circ$ :
$\frac{14.2 \sin 34^\circ}{\sin 102^\circ} = 8.12\ldots$
Find AB, which is opposite angle $44^\circ$ :
$\frac{14.2 \sin 44^\circ}{\sin 102^\circ} = 10.08\ldots$
Add the three sides:
$8.12\ldots + 10.08\ldots = 32.40\ldots$
To 3 significant figures, the perimeter is 32.4 m.

Area questions using the sine rule

For a triangle, the area formula using two sides and the included angle is:

Area=12absin⁡C\text{Area} = \frac{1}{2}ab\sin C

The included angle is the angle between the two sides you are using.

Example

Using sine rule before area

Triangle ABC has AB = 10 m, angle $A$ is 76°, and angle $C$ is 44°. Find the area to 1 decimal place.

Find angle $B$ :
$180^\circ - 76^\circ - 44^\circ = 60^\circ$
AB is opposite angle $44^\circ$ . To use the area formula with angle $A$ , find AC, which is opposite angle $60^\circ$ :
$\frac{10 \sin 60^\circ}{\sin 44^\circ} = 12.47\ldots$
Use the area formula with sides AB and AC, and included angle $76^\circ$ :
$Area=12×10×12.47…×sin⁡76∘\text{Area} = \frac{1}{2} \times 10 \times 12.47\ldots \times \sin 76^\circ$
Calculate and round:
$Area=60.5…\text{Area} = 60.5\ldots$
To 1 decimal place, the area is 60.5 m².

The obtuse angle issue

When you use inverse sine, your calculator gives the acute angle first. But sine has the same value for an angle and its supplement:

sin⁡θ=sin⁡(180∘−θ)\sin \theta = \sin(180^\circ - \theta)

This matters when a question says an angle is obtuse, meaning it is greater than 90° and less than 180°.

A diagram showing an acute angle and its obtuse supplement having the same sine value

Common Mistake

Check for an obtuse angle

If the question says the angle is obtuse, do not stop at the calculator’s acute answer. Subtract it from 180°.

Example

Finding an obtuse angle

Triangle ABC has AC = 18 cm, AB = 11 cm, and angle $C$ is 32°. Angle $B$ is obtuse. Find angle $B$ to 3 significant figures.

Match the opposite pairs:
- AB = 11 cm is opposite angle $32^\circ$ .
- AC = 18 cm is opposite angle $B$ .
Set up the sine rule:
$sin⁡B18=sin⁡32∘11\frac{\sin B}{18} = \frac{\sin 32^\circ}{11}$
Multiply both sides by 18:
$sin⁡B=18sin⁡32∘11\sin B = \frac{18 \sin 32^\circ}{11}$
Calculate:
$sin⁡B=0.867…\sin B = 0.867\ldots$
Use inverse sine to get the acute angle:
$sin⁡−1(0.867…)=60.1…∘\sin^{-1}(0.867\ldots) = 60.1\ldots^\circ$
Since angle $B$ is obtuse, subtract from 180°:
$180^\circ - 60.1\ldots^\circ = 119.8\ldots^\circ$
To 3 significant figures, $120^\circ$ .

Exam technique

In the exam

Mark the opposite pairs clearly before writing the formula.
If finding a side, use $sidesin⁡(angle)\frac{\text{side}}{\sin(\text{angle})}$ . If finding an angle, use $sin⁡(angle)side\frac{\sin(\text{angle})}{\text{side}}$ .
Check whether the question asks for decimal places or significant figures, and only round at the end.
If using $sin^{-1}$ , check whether an obtuse angle is required.

Self review

Check yourself

Can you identify the side opposite a given angle without guessing?
Do you know which version of the sine rule is easier for finding a side?
If your calculator gives an acute angle, when might you need to subtract from 180°?

The Sine Rule

What you'll learn

When the sine rule is useful for non-right-angled triangles.
How to match each side with its opposite angle.
How to find missing sides and missing angles using the sine rule.
How to combine the sine rule with perimeter and area questions.

Before you start: the triangle facts you need

The sine rule is used in triangles that are not necessarily right-angled. You do not need a 90° angle, and you usually cannot use SOHCAHTOA directly.

The most important skill is spotting opposite pairs: an angle and the side directly across from it.

A labelled triangle showing angles A, B, C and opposite sides a, b, c

Definition

Opposite side

You will also use the angle sum of a triangle:

180^\circ

Example

Finding the missing angle and an opposite pair

A triangle has angles 37° and 98°. A side of length 9 cm is opposite the 98° angle. Find the third angle and identify the side-angle pair.

Use the fact that angles in a triangle add to 180°:
$37∘+98∘=135∘37^\circ + 98^\circ = 135^\circ$
Subtract from 180°:
$180∘−135∘=45∘180^\circ - 135^\circ = 45^\circ$
The third angle is 45°.
The side of length 9 cm is paired with the 98° angle, because they are opposite each other.

Common Mistake

Do not pair an angle with a touching side

The sine rule only works with opposite pairs. If an angle touches a side, that does not mean they are paired for the sine rule.

The sine rule

The sine rule links the sides and angles of any triangle.

Definition

The sine rule

For triangle ABC, where sides $a$ , $b$ , and $c$ are opposite angles $A$ , $B$ , and $C$ :

asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

You can also write it as:

sin⁡Aa=sin⁡Bb=sin⁡Cc\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}

Key Idea

The big idea

Use the sine rule when you know one complete opposite pair, and you want another side or angle from a second opposite pair.

A complete opposite pair means you know both the side and the angle opposite it.

Finding a missing side

When finding a side, it is usually easiest to put the side lengths on top:

angle)\frac{\text{side}}{\sin(\text{opposite angle})}

You need:

one known side with its opposite angle;
the angle opposite the side you want.

Example

Finding a missing length

In a triangle, a side of length 12 cm is opposite an angle of 44°. Another side, $y$ cm, is opposite an angle of 101°. Find $y$ to 1 decimal place.

Match the opposite pairs:
- 12 cm is opposite 44°.
- $y$ cm is opposite 101°.
Set up the sine rule with sides on top:
$ysin⁡101∘=12sin⁡44∘\frac{y}{\sin 101^\circ} = \frac{12}{\sin 44^\circ}$
Multiply both sides by $sin⁡101∘\sin 101^\circ$ :
$\frac{12 \sin 101^\circ}{\sin 44^\circ}$
Calculate:
$16.96\ldots$
Round to 1 decimal place: $y = 17.0$ cm.

Tip

Calculator mode

Make sure your calculator is in degrees mode, not radians. In degrees mode, $sin⁡44∘\sin 44^\circ$ means the sine of 44°.

Finding a missing angle

When finding an angle, it is usually easier to put the sine parts on top:

sin⁡Aa=sin⁡Bb\frac{\sin A}{a} = \frac{\sin B}{b}

After finding $sin⁡A\sin A$ , use the inverse sine button on your calculator. This is often written as $sin^{-1}$ .

Definition

Inverse sine

The inverse sine, $sin^{-1}$ , finds the angle when you already know its sine value.

Example

Finding a missing angle

In a triangle, a side of length 8.1 cm is opposite angle $x$ . A side of length 12.6 cm is opposite an angle of 72°. Find $x$ to 3 significant figures.

Match the opposite pairs:
- 8.1 cm is opposite angle $x$ .
- 12.6 cm is opposite 72°.
Use the sine rule with sine parts on top:
$sin⁡x8.1=sin⁡72∘12.6\frac{\sin x}{8.1} = \frac{\sin 72^\circ}{12.6}$
Multiply both sides by 8.1:
$sin⁡x=8.1sin⁡72∘12.6\sin x = \frac{8.1 \sin 72^\circ}{12.6}$
Calculate the sine value:
$sin⁡x=0.611…\sin x = 0.611\ldots$
Use inverse sine:
$\sin^{-1}(0.611\ldots)$
Round to 3 significant figures: $37.7^\circ$ .

Common Mistake

Rounding too early

Do not round the sine value halfway through the question. Keep the full calculator value until the final answer, then round as requested.

Sometimes you must find a missing angle first

Some questions give two angles and one side. If the angle opposite the known side is not directly given, find the third angle first.

Example

Using the angle sum before the sine rule

In triangle ABC, angle $A$ is 36°, angle $B$ is 82°, and side AB is 7 m. Find side AC to 3 significant figures.

Work out angle $C$ because side AB is opposite angle $C$ :
$180^\circ - 36^\circ - 82^\circ = 62^\circ$
Match the pairs:
- AB = 7 m is opposite angle $62^\circ$ .
- AC is opposite angle $82^\circ$ .
Use the sine rule:
$ACsin⁡82∘=7sin⁡62∘\frac{AC}{\sin 82^\circ} = \frac{7}{\sin 62^\circ}$
Rearrange and calculate:
$\frac{7 \sin 82^\circ}{\sin 62^\circ} = 7.85\ldots$
To 3 significant figures, AC = 7.85 m.

Perimeter questions

A perimeter is the total distance around the outside of a shape. For a triangle, add the three side lengths.

In sine rule perimeter questions, you may need to find two missing sides first.

Example

Finding the perimeter of a triangle

Triangle ABC has AC = 14.2 m, angle $B$ is 102°, and angle $C$ is 44°. Find the perimeter to 3 significant figures.

First find angle $A$ :
$180^\circ - 102^\circ - 44^\circ = 34^\circ$
Match the known pair:
- AC = 14.2 m is opposite angle $102^\circ$ .
Find BC, which is opposite angle $34^\circ$ :
$\frac{14.2 \sin 34^\circ}{\sin 102^\circ} = 8.12\ldots$
Find AB, which is opposite angle $44^\circ$ :
$\frac{14.2 \sin 44^\circ}{\sin 102^\circ} = 10.08\ldots$
Add the three sides:
$8.12\ldots + 10.08\ldots = 32.40\ldots$
To 3 significant figures, the perimeter is 32.4 m.

Area questions using the sine rule

For a triangle, the area formula using two sides and the included angle is:

Area=12absin⁡C\text{Area} = \frac{1}{2}ab\sin C

The included angle is the angle between the two sides you are using.

Example

Using sine rule before area

Triangle ABC has AB = 10 m, angle $A$ is 76°, and angle $C$ is 44°. Find the area to 1 decimal place.

Find angle $B$ :
$180^\circ - 76^\circ - 44^\circ = 60^\circ$
AB is opposite angle $44^\circ$ . To use the area formula with angle $A$ , find AC, which is opposite angle $60^\circ$ :
$\frac{10 \sin 60^\circ}{\sin 44^\circ} = 12.47\ldots$
Use the area formula with sides AB and AC, and included angle $76^\circ$ :
$Area=12×10×12.47…×sin⁡76∘\text{Area} = \frac{1}{2} \times 10 \times 12.47\ldots \times \sin 76^\circ$
Calculate and round:
$Area=60.5…\text{Area} = 60.5\ldots$
To 1 decimal place, the area is 60.5 m².

The obtuse angle issue

When you use inverse sine, your calculator gives the acute angle first. But sine has the same value for an angle and its supplement:

sin⁡θ=sin⁡(180∘−θ)\sin \theta = \sin(180^\circ - \theta)

This matters when a question says an angle is obtuse, meaning it is greater than 90° and less than 180°.

A diagram showing an acute angle and its obtuse supplement having the same sine value

Common Mistake

Check for an obtuse angle

If the question says the angle is obtuse, do not stop at the calculator’s acute answer. Subtract it from 180°.

Example

Finding an obtuse angle

Triangle ABC has AC = 18 cm, AB = 11 cm, and angle $C$ is 32°. Angle $B$ is obtuse. Find angle $B$ to 3 significant figures.

Match the opposite pairs:
- AB = 11 cm is opposite angle $32^\circ$ .
- AC = 18 cm is opposite angle $B$ .
Set up the sine rule:
$sin⁡B18=sin⁡32∘11\frac{\sin B}{18} = \frac{\sin 32^\circ}{11}$
Multiply both sides by 18:
$sin⁡B=18sin⁡32∘11\sin B = \frac{18 \sin 32^\circ}{11}$
Calculate:
$sin⁡B=0.867…\sin B = 0.867\ldots$
Use inverse sine to get the acute angle:
$sin⁡−1(0.867…)=60.1…∘\sin^{-1}(0.867\ldots) = 60.1\ldots^\circ$
Since angle $B$ is obtuse, subtract from 180°:
$180^\circ - 60.1\ldots^\circ = 119.8\ldots^\circ$
To 3 significant figures, $120^\circ$ .

Exam technique

In the exam

Mark the opposite pairs clearly before writing the formula.
If finding a side, use $sidesin⁡(angle)\frac{\text{side}}{\sin(\text{angle})}$ . If finding an angle, use $sin⁡(angle)side\frac{\sin(\text{angle})}{\text{side}}$ .
Check whether the question asks for decimal places or significant figures, and only round at the end.
If using $sin^{-1}$ , check whether an obtuse angle is required.

Self review

Check yourself

Can you identify the side opposite a given angle without guessing?
Do you know which version of the sine rule is easier for finding a side?
If your calculator gives an acute angle, when might you need to subtract from 180°?

The Sine Rule

What you'll learn

Before you start: the triangle facts you need

The sine rule

Finding a missing side

Finding a missing angle

Sometimes you must find a missing angle first

Perimeter questions

Area questions using the sine rule

The obtuse angle issue

Recap questions

You've reached the end

The Sine Rule

What you'll learn

Before you start: the triangle facts you need

The sine rule

Finding a missing side

Finding a missing angle

Sometimes you must find a missing angle first

Perimeter questions

Area questions using the sine rule

The obtuse angle issue

Recap questions

You've reached the end

The Sine Rule

What you'll learn

Before you start: the triangle facts you need

The sine rule

Finding a missing side

Finding a missing angle

Sometimes you must find a missing angle first

Perimeter questions

Area questions using the sine rule

The obtuse angle issue

Recap questions

You've reached the end

The Sine Rule Revision Guide

The Sine Rule

What you'll learn

Before you start: the triangle facts you need

The sine rule

Finding a missing side

Finding a missing angle

Sometimes you must find a missing angle first

Perimeter questions

Area questions using the sine rule

The obtuse angle issue

Recap questions

You've reached the end

The Sine Rule Revision Guide