The Cosine Rule

What you'll learn

How to spot when the Cosine Rule is needed.
How to find a missing side when you know two sides and the included angle.
How to find a missing angle when you know all three sides.
How the Cosine Rule can combine with algebra or the triangle area formula.

Before you start: the triangle language

The Cosine Rule works in any triangle, not just right-angled triangles. It is especially useful when Pythagoras and basic trigonometry do not apply directly.

Definition

Included angle and opposite side

The included angle is the angle between two known sides. The opposite side is the side directly across from an angle.

In the diagram below, angle $C$ is between sides $a$ and $b$ , and side $c$ is opposite angle $C$ .

Triangle showing side and angle labels for the Cosine Rule

Example

Spotting the included angle

Suppose a triangle has sides 8 cm and 11 cm with a 70° angle between them.
The 70° angle is the included angle, because it sits between the two known sides.
The side opposite the 70° angle is the side you can find directly using the Cosine Rule.

Tip

Quick check

If the angle is drawn between the two given sides, think Cosine Rule for a side.

The Cosine Rule for finding a missing side

Definition

The Cosine Rule

If side $c$ is opposite angle $C$ , and the two sides around angle $C$ are $a$ and $b$ , then:

c^2 = a^2 + b^2 - 2ab\cos C

This looks a bit like Pythagoras, but with an extra cosine part. In fact, when $90^\circ$ , $cos⁡90∘=0\cos 90^\circ = 0$ , so it becomes Pythagoras.

Key Idea

Main idea

Use $c^2 = a^2 + b^2 - 2ab\cos C$ when you know two sides and the angle between them, and you want the third side.

Example

Finding a missing side

A triangle has two sides of length 12 cm and 9 cm. The angle between them is 105°. Find the opposite side $x$ , to 1 decimal place.

Identify the two known sides and the included angle: the sides are 12 cm and 9 cm, and the included angle is 105°.
Substitute into the Cosine Rule:
$x2=122+92−2×12×9×cos⁡105∘x^2 = 12^2 + 9^2 - 2 \times 12 \times 9 \times \cos 105^\circ$
Work out the value of $x^2$ :
$x2≈280.9x^2 \approx 280.9$
Square root and round to 1 decimal place: $\approx 16.8$ cm.

Common Mistake

Forgetting the square root

The formula gives $x^2$ first, not $x$ . Always square root at the end when finding a length.

Tip

Obtuse angle sanity check

If the included angle is bigger than 90°, the opposite side is usually quite long. That is because the triangle is opening out wider.

The Cosine Rule for finding a missing angle

Sometimes you are given all three sides and asked for an angle. Then you rearrange the Cosine Rule.

Starting from:

c^2 = a^2 + b^2 - 2ab\cos C

Rearrange to get:

cos⁡C=a2+b2−c22ab\cos C = \frac{a^2 + b^2 - c^2}{2ab}

Then use inverse cosine, written $cos^{-1}$ , to get the angle.

Definition

Inverse cosine

The inverse cosine, $cos^{-1}$ , is the calculator function that finds the angle when you know its cosine value.

Example

Finding an angle from three sides

In triangle $A B C$ , $A B = 6$ m, $B C = 8$ m and $A C = 9$ m. Find angle $A B C$ to the nearest degree.

Angle $A B C$ is at $B$ , so the two sides touching the angle are $A B = 6$ and $B C = 8$ .
The side opposite angle $B$ is $A C = 9$ .
Substitute into the rearranged Cosine Rule:
$cos⁡B=62+82−922×6×8\cos B = \frac{6^2 + 8^2 - 9^2}{2 \times 6 \times 8}$
Simplify the fraction:
$cos⁡B=1996\cos B = \frac{19}{96}$
Use inverse cosine:
$\cos^{-1}\left(\frac{19}{96}\right) \approx 78.6^\circ$
Round to the nearest degree: angle $A B C$ is 79°.

Common Mistake

Using the wrong opposite side

When finding an angle, the side opposite the angle is the one being subtracted in the numerator: $a^2 + b^2 - c^2$ . Label the opposite side before substituting.

Tip

Calculator mode

Make sure your calculator is in degrees, not radians. Your answer should be in degrees for these questions.

Choosing between the two versions

Use the Cosine Rule when you have either:

two sides and the included angle, and you need the third side;
all three sides, and you need an angle.

If you have a right-angled triangle, try Pythagoras or SOHCAHTOA first. If you have a matching opposite side and angle pair, the Sine Rule may be more suitable.

When side lengths contain algebra

Some harder questions give side lengths like $x + 1$ or $2 x - 1$ . You still use the Cosine Rule, but you will need to expand brackets and solve an equation.

Definition

Quadratic equation

A quadratic equation is an equation involving $x^2$ , such as $x^2 - x - 20 = 0$ .

Example

Using the Cosine Rule with algebra

A triangle has sides $x + 1$ and $2 x - 1$ with an included angle of 60°. The opposite side is $63\sqrt{63}$ . Find $x$ .

Since the opposite side is $63\sqrt{63}$ , its square is 63.
Substitute into the Cosine Rule:
$1)^2 + (2x - 1)^2 - 2(x + 1)(2x - 1)\cos 60^\circ$
Use $cos⁡60∘=12\cos 60^\circ = \frac{1}{2}$ :
$63 = (x + 1)^2 + (2x - 1)^2 - (x + 1)(2x - 1)$
Expand and simplify:
$63 = 3x^2 - 3x + 3$
Rearrange and factorise:
$3x2−3x−60=0⇒x2−x−20=0⇒(x−5)(x+4)=03x^2 - 3x - 60 = 0 \Rightarrow x^2 - x - 20 = 0 \Rightarrow (x - 5)(x + 4) = 0$
The solutions are $x = 5$ or $x = - 4$ . Reject $x = - 4$ because it would give negative side lengths, so $x = 5$ .

Common Mistake

Accepting an impossible value

Lengths cannot be negative. If an algebra question gives two possible values, check that all side lengths are positive.

Combining area and the Cosine Rule

You may need to use the triangle area formula first.

Definition

Area of a triangle using sine

If two sides are $a$ and $b$ with included angle $C$ , then the area is:

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

This can help you find the included angle. Then you can use the Cosine Rule to find the missing side and finish the perimeter.

Example

Finding a perimeter using area first

Triangle $A B C$ has $A B = 18.5$ m, $A C = 12.8$ m, and area 70 m². The angle at $A$ is acute. Find the perimeter to 3 significant figures.

Use the area formula with the two sides around angle $A$ :
$\frac{1}{2} \times 18.5 \times 12.8 \times \sin A$
Rearrange to find $sin⁡A\sin A$ :
$sin⁡A=14018.5×12.8≈0.591\sin A = \frac{140}{18.5 \times 12.8} \approx 0.591$
Use inverse sine, choosing the acute angle:
$\approx \sin^{-1}(0.591) \approx 36.2^\circ$
Use the Cosine Rule to find $B C$ :
$BC2=18.52+12.82−2×18.5×12.8×cos⁡36.2∘BC^2 = 18.5^2 + 12.8^2 - 2 \times 18.5 \times 12.8 \times \cos 36.2^\circ$
This gives $\approx 11.1$ m, so the perimeter is approximately 42.4 m to 3 significant figures.

Common Mistake

Sine ambiguity

When you use inverse sine, there may be two possible angles between 0° and 180°. Use the diagram or wording, such as “acute” or “obtuse”, to choose the correct one.

Rounding answers

A question may ask for:

1 decimal place, meaning one digit after the decimal point;
3 significant figures, meaning the first three important digits.

Keep full calculator values until the final line. Rounding too early can change the final answer.

Exam technique

In the exam

Mark the angle you are using, then mark the side opposite it.
Write the Cosine Rule formula before substituting numbers, so the method is clear.
Do not round too early; give the final answer in the form requested by the question.

Self review

Check yourself

If you know two sides and the angle between them, which version of the Cosine Rule do you use?
When finding an angle, which side goes in the “minus” part of the numerator?
Why must you check side lengths when solving an algebraic Cosine Rule question?

The Cosine Rule

What you'll learn

How to spot when the Cosine Rule is needed.
How to find a missing side when you know two sides and the included angle.
How to find a missing angle when you know all three sides.
How the Cosine Rule can combine with algebra or the triangle area formula.

Before you start: the triangle language

The Cosine Rule works in any triangle, not just right-angled triangles. It is especially useful when Pythagoras and basic trigonometry do not apply directly.

Definition

Included angle and opposite side

The included angle is the angle between two known sides. The opposite side is the side directly across from an angle.

In the diagram below, angle $C$ is between sides $a$ and $b$ , and side $c$ is opposite angle $C$ .

Triangle showing side and angle labels for the Cosine Rule

Example

Spotting the included angle

Suppose a triangle has sides 8 cm and 11 cm with a 70° angle between them.
The 70° angle is the included angle, because it sits between the two known sides.
The side opposite the 70° angle is the side you can find directly using the Cosine Rule.

Tip

Quick check

If the angle is drawn between the two given sides, think Cosine Rule for a side.

The Cosine Rule for finding a missing side

Definition

The Cosine Rule

If side $c$ is opposite angle $C$ , and the two sides around angle $C$ are $a$ and $b$ , then:

c^2 = a^2 + b^2 - 2ab\cos C

This looks a bit like Pythagoras, but with an extra cosine part. In fact, when $90^\circ$ , $cos⁡90∘=0\cos 90^\circ = 0$ , so it becomes Pythagoras.

Key Idea

Main idea

Use $c^2 = a^2 + b^2 - 2ab\cos C$ when you know two sides and the angle between them, and you want the third side.

Example

Finding a missing side

A triangle has two sides of length 12 cm and 9 cm. The angle between them is 105°. Find the opposite side $x$ , to 1 decimal place.

Identify the two known sides and the included angle: the sides are 12 cm and 9 cm, and the included angle is 105°.
Substitute into the Cosine Rule:
$x2=122+92−2×12×9×cos⁡105∘x^2 = 12^2 + 9^2 - 2 \times 12 \times 9 \times \cos 105^\circ$
Work out the value of $x^2$ :
$x2≈280.9x^2 \approx 280.9$
Square root and round to 1 decimal place: $\approx 16.8$ cm.

Common Mistake

Forgetting the square root

The formula gives $x^2$ first, not $x$ . Always square root at the end when finding a length.

Tip

Obtuse angle sanity check

If the included angle is bigger than 90°, the opposite side is usually quite long. That is because the triangle is opening out wider.

The Cosine Rule for finding a missing angle

Sometimes you are given all three sides and asked for an angle. Then you rearrange the Cosine Rule.

Starting from:

c^2 = a^2 + b^2 - 2ab\cos C

Rearrange to get:

cos⁡C=a2+b2−c22ab\cos C = \frac{a^2 + b^2 - c^2}{2ab}

Then use inverse cosine, written $cos^{-1}$ , to get the angle.

Definition

Inverse cosine

The inverse cosine, $cos^{-1}$ , is the calculator function that finds the angle when you know its cosine value.

Example

Finding an angle from three sides

In triangle $A B C$ , $A B = 6$ m, $B C = 8$ m and $A C = 9$ m. Find angle $A B C$ to the nearest degree.

Angle $A B C$ is at $B$ , so the two sides touching the angle are $A B = 6$ and $B C = 8$ .
The side opposite angle $B$ is $A C = 9$ .
Substitute into the rearranged Cosine Rule:
$cos⁡B=62+82−922×6×8\cos B = \frac{6^2 + 8^2 - 9^2}{2 \times 6 \times 8}$
Simplify the fraction:
$cos⁡B=1996\cos B = \frac{19}{96}$
Use inverse cosine:
$\cos^{-1}\left(\frac{19}{96}\right) \approx 78.6^\circ$
Round to the nearest degree: angle $A B C$ is 79°.

Common Mistake

Using the wrong opposite side

When finding an angle, the side opposite the angle is the one being subtracted in the numerator: $a^2 + b^2 - c^2$ . Label the opposite side before substituting.

Tip

Calculator mode

Make sure your calculator is in degrees, not radians. Your answer should be in degrees for these questions.

Choosing between the two versions

Use the Cosine Rule when you have either:

two sides and the included angle, and you need the third side;
all three sides, and you need an angle.

If you have a right-angled triangle, try Pythagoras or SOHCAHTOA first. If you have a matching opposite side and angle pair, the Sine Rule may be more suitable.

When side lengths contain algebra

Some harder questions give side lengths like $x + 1$ or $2 x - 1$ . You still use the Cosine Rule, but you will need to expand brackets and solve an equation.

Definition

Quadratic equation

A quadratic equation is an equation involving $x^2$ , such as $x^2 - x - 20 = 0$ .

Example

Using the Cosine Rule with algebra

A triangle has sides $x + 1$ and $2 x - 1$ with an included angle of 60°. The opposite side is $63\sqrt{63}$ . Find $x$ .

Since the opposite side is $63\sqrt{63}$ , its square is 63.
Substitute into the Cosine Rule:
$1)^2 + (2x - 1)^2 - 2(x + 1)(2x - 1)\cos 60^\circ$
Use $cos⁡60∘=12\cos 60^\circ = \frac{1}{2}$ :
$63 = (x + 1)^2 + (2x - 1)^2 - (x + 1)(2x - 1)$
Expand and simplify:
$63 = 3x^2 - 3x + 3$
Rearrange and factorise:
$3x2−3x−60=0⇒x2−x−20=0⇒(x−5)(x+4)=03x^2 - 3x - 60 = 0 \Rightarrow x^2 - x - 20 = 0 \Rightarrow (x - 5)(x + 4) = 0$
The solutions are $x = 5$ or $x = - 4$ . Reject $x = - 4$ because it would give negative side lengths, so $x = 5$ .

Common Mistake

Accepting an impossible value

Lengths cannot be negative. If an algebra question gives two possible values, check that all side lengths are positive.

Combining area and the Cosine Rule

You may need to use the triangle area formula first.

Definition

Area of a triangle using sine

If two sides are $a$ and $b$ with included angle $C$ , then the area is:

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

This can help you find the included angle. Then you can use the Cosine Rule to find the missing side and finish the perimeter.

Example

Finding a perimeter using area first

Triangle $A B C$ has $A B = 18.5$ m, $A C = 12.8$ m, and area 70 m². The angle at $A$ is acute. Find the perimeter to 3 significant figures.

Use the area formula with the two sides around angle $A$ :
$\frac{1}{2} \times 18.5 \times 12.8 \times \sin A$
Rearrange to find $sin⁡A\sin A$ :
$sin⁡A=14018.5×12.8≈0.591\sin A = \frac{140}{18.5 \times 12.8} \approx 0.591$
Use inverse sine, choosing the acute angle:
$\approx \sin^{-1}(0.591) \approx 36.2^\circ$
Use the Cosine Rule to find $B C$ :
$BC2=18.52+12.82−2×18.5×12.8×cos⁡36.2∘BC^2 = 18.5^2 + 12.8^2 - 2 \times 18.5 \times 12.8 \times \cos 36.2^\circ$
This gives $\approx 11.1$ m, so the perimeter is approximately 42.4 m to 3 significant figures.

Common Mistake

Sine ambiguity

When you use inverse sine, there may be two possible angles between 0° and 180°. Use the diagram or wording, such as “acute” or “obtuse”, to choose the correct one.

Rounding answers

A question may ask for:

1 decimal place, meaning one digit after the decimal point;
3 significant figures, meaning the first three important digits.

Keep full calculator values until the final line. Rounding too early can change the final answer.

Exam technique

In the exam

Mark the angle you are using, then mark the side opposite it.
Write the Cosine Rule formula before substituting numbers, so the method is clear.
Do not round too early; give the final answer in the form requested by the question.

Self review

Check yourself

If you know two sides and the angle between them, which version of the Cosine Rule do you use?
When finding an angle, which side goes in the “minus” part of the numerator?
Why must you check side lengths when solving an algebraic Cosine Rule question?

The Cosine Rule

What you'll learn

Before you start: the triangle language

The Cosine Rule for finding a missing side

The Cosine Rule for finding a missing angle

Choosing between the two versions

When side lengths contain algebra

Combining area and the Cosine Rule

Rounding answers

Recap questions

You've reached the end

The Cosine Rule

What you'll learn

Before you start: the triangle language

The Cosine Rule for finding a missing side

The Cosine Rule for finding a missing angle

Choosing between the two versions

When side lengths contain algebra

Combining area and the Cosine Rule

Rounding answers

Recap questions

You've reached the end

The Cosine Rule

What you'll learn

Before you start: the triangle language

The Cosine Rule for finding a missing side

The Cosine Rule for finding a missing angle

Choosing between the two versions

When side lengths contain algebra

Combining area and the Cosine Rule

Rounding answers

Recap questions

You've reached the end

The Cosine Rule Revision Guide

The Cosine Rule

What you'll learn

Before you start: the triangle language

The Cosine Rule for finding a missing side

The Cosine Rule for finding a missing angle

Choosing between the two versions

When side lengths contain algebra

Combining area and the Cosine Rule

Rounding answers

Recap questions

You've reached the end

The Cosine Rule Revision Guide