Exact trig values

What you'll learn

What “exact” means in trigonometry.
The exact values of sine, cosine and tangent for 0°, 30°, 45°, 60° and 90°.
How to use special triangles to remember the values.
How to use exact trig values to find missing lengths in right-angled triangles.

What does “exact” mean?

In this topic, you are not estimating with a calculator. You are giving the precise value.

Definition

Exact value

An exact value is an answer written without rounding. It might be a whole number, a fraction, or a surd, which is a square root that does not simplify to a whole number, such as $2\sqrt{2}$ .

For example, 0.707 is only an approximation. The exact value of $sin⁡(45∘)\sin(45^\circ)$ is $22\frac{\sqrt{2}}{2}$ .

Example

Exact, not rounded

Suppose you are asked for $sin⁡(45∘)\sin(45^\circ)$ .
A calculator gives a decimal starting 0.7071..., but this is rounded.
The exact GCSE answer is:
$sin⁡(45∘)=22\sin(45^\circ)=\frac{\sqrt{2}}{2}$

Common Mistake

Using decimals too early

If the question asks for an exact value, do not write a rounded decimal like 0.866. Use the fraction or surd form instead.

Right-angled triangle basics

Exact trig values come from special right-angled triangles.

Definition

The three side names

The hypotenuse is the longest side of a right-angled triangle. It is opposite the right angle.
The opposite side is opposite the angle you are using.
The adjacent side is next to the angle you are using, but it is not the hypotenuse.

The hypotenuse is opposite the right angle, while opposite and adjacent are named relative to the chosen angle θ.

We often call the angle we are using $θ\theta$ , pronounced “theta”.

Key Idea

SOHCAHTOA

SOHCAHTOA reminds you that $sin⁡θ=oppositehypotenuse\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$ , $cos⁡θ=adjacenthypotenuse\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$ and $tan⁡θ=oppositeadjacent\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$ .

Example

Choosing the correct trig ratio

Imagine a right-angled triangle where the side opposite $θ\theta$ is 6 cm and the hypotenuse is 12 cm.

For this triangle, the given sides are opposite θ and hypotenuse, so the sine ratio is used.

Opposite and hypotenuse means use sine:
$sin⁡θ=oppositehypotenuse\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$
Substitute the side lengths:
$sin⁡θ=612\sin\theta=\frac{6}{12}$
Simplify:
$sin⁡θ=12\sin\theta=\frac{1}{2}$

The 45° special triangle

An isosceles triangle has two equal sides. If you make an isosceles right-angled triangle, the two smaller angles are both 45°.

The useful side lengths are:

The 45° special triangle has equal shorter sides 1 and 1, with hypotenuse √2.

two shorter sides: 1 and 1
hypotenuse: $2\sqrt{2}$

So for a 45° angle, the opposite and adjacent sides are the same.

Example

Finding an exact value at 45°

Use the 45-45-90 triangle with shorter sides 1 and 1, and hypotenuse $2\sqrt{2}$ .

Using either 45° angle, the opposite side is 1 and the hypotenuse is √2.

For $sin⁡(45∘)\sin(45^\circ)$ , use opposite over hypotenuse:
$sin⁡(45∘)=12\sin(45^\circ)=\frac{1}{\sqrt{2}}$
Remove the square root from the denominator:
$12=22\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$
Therefore:
$sin⁡(45∘)=22\sin(45^\circ)=\frac{\sqrt{2}}{2}$

Tip

45° shortcut

For 45°, sine and cosine are equal: $sin⁡(45∘)=cos⁡(45∘)=22\sin(45^\circ)=\cos(45^\circ)=\frac{\sqrt{2}}{2}$ .

The 30° and 60° special triangle

An equilateral triangle has all three sides equal and all three angles 60°. If you cut one in half, you get a 30-60-90 right-angled triangle.

Cutting an equilateral triangle in half creates a 30°–60°–90° right-angled triangle.

The useful side lengths are:

shortest side: 1
middle side: $3\sqrt{3}$
hypotenuse: 2

For the 30° angle, the opposite side is the shortest side.
For the 60° angle, the opposite side is the $3\sqrt{3}$ side.

In the 30°–60°–90° triangle, the side opposite 60° is √3 and the side opposite 30° is 1.

Example

Finding an exact value at 60°

Use the 30-60-90 triangle with side lengths 1, $3\sqrt{3}$ and 2.
For a 60° angle, the opposite side is $3\sqrt{3}$ and the adjacent side is 1.
Tangent uses opposite over adjacent:
$tan⁡(60∘)=31\tan(60^\circ)=\frac{\sqrt{3}}{1}$
So $tan⁡(60∘)=3\tan(60^\circ)=\sqrt{3}$ .

Common Mistake

Swapping 30° and 60°

For sine, 30° has the smaller value $12\frac{1}{2}$ and 60° has the larger value $32\frac{\sqrt{3}}{2}$ . Cosine is the other way round.

The exact values to know

Key Idea

Memorise these values

0°: $sin⁡(0∘)=0\sin(0^\circ)=0$ , $cos⁡(0∘)=1\cos(0^\circ)=1$ , $tan⁡(0∘)=0\tan(0^\circ)=0$
30°: $sin⁡(30∘)=12\sin(30^\circ)=\frac{1}{2}$ , $cos⁡(30∘)=32\cos(30^\circ)=\frac{\sqrt{3}}{2}$ , $tan⁡(30∘)=33\tan(30^\circ)=\frac{\sqrt{3}}{3}$
45°: $sin⁡(45∘)=22\sin(45^\circ)=\frac{\sqrt{2}}{2}$ , $cos⁡(45∘)=22\cos(45^\circ)=\frac{\sqrt{2}}{2}$ , $tan⁡(45∘)=1\tan(45^\circ)=1$
60°: $sin⁡(60∘)=32\sin(60^\circ)=\frac{\sqrt{3}}{2}$ , $cos⁡(60∘)=12\cos(60^\circ)=\frac{1}{2}$ , $tan⁡(60∘)=3\tan(60^\circ)=\sqrt{3}$
90°: $sin⁡(90∘)=1\sin(90^\circ)=1$ , $cos⁡(90∘)=0\cos(90^\circ)=0$

Common Mistake

Why tan 90° is missing

$tan⁡(90∘)\tan(90^\circ)$ is not defined because tangent involves dividing by the adjacent side, and at 90° this would mean dividing by zero.

Example

Writing down exact values

For $cos⁡(90∘)\cos(90^\circ)$ , look at the 90° values: $cos⁡(90∘)=0\cos(90^\circ)=0$ .
For $tan⁡(30∘)\tan(30^\circ)$ , look at the 30° values: $tan⁡(30∘)=33\tan(30^\circ)=\frac{\sqrt{3}}{3}$ .
For $sin⁡(60∘)\sin(60^\circ)$ , look at the 60° values: $sin⁡(60∘)=32\sin(60^\circ)=\frac{\sqrt{3}}{2}$ .

Tip

Sine pattern

The sine values from 0° to 90° go up as $02\frac{\sqrt{0}}{2}$ , $12\frac{\sqrt{1}}{2}$ , $22\frac{\sqrt{2}}{2}$ , $32\frac{\sqrt{3}}{2}$ , $42\frac{\sqrt{4}}{2}$ . Cosine uses the same pattern backwards.

Using exact trig values to find lengths

You may also be given a right-angled triangle and asked to calculate a missing side. The method is the same as normal trigonometry, but you substitute an exact value instead of a calculator decimal.

Example

Finding a hypotenuse using 30°

A right-angled triangle has a 30° angle. The side opposite this angle is 18 cm. Let the hypotenuse be $h$ .

The side opposite the 30° angle is given as 18 cm and the hypotenuse is h.

Opposite and hypotenuse means use sine:
$sin⁡(30∘)=18h\sin(30^\circ)=\frac{18}{h}$
Substitute the exact value $sin⁡(30∘)=12\sin(30^\circ)=\frac{1}{2}$ :
$12=18h\frac{1}{2}=\frac{18}{h}$
Solve for $h$ :
$h = 36$
The hypotenuse is 36 cm.

Example

Finding an adjacent side using 60°

A right-angled triangle has a 60° angle. The side opposite this angle is 18 cm. Let the adjacent side be $a$ .

The side opposite the 60° angle is 18 cm and the adjacent side to find is a.

Opposite and adjacent means use tangent:
$tan⁡(60∘)=18a\tan(60^\circ)=\frac{18}{a}$
Substitute the exact value $tan⁡(60∘)=3\tan(60^\circ)=\sqrt{3}$ :
$3=18a\sqrt{3}=\frac{18}{a}$
Rearrange and simplify:
$a=183=1833=63a=\frac{18}{\sqrt{3}}=\frac{18\sqrt{3}}{3}=6\sqrt{3}$
The adjacent side is $636\sqrt{3}$ cm.

Exam technique

In the exam

First identify the angle: is it 0°, 30°, 45°, 60° or 90°?
If it is a “write down” question, use the memorised exact value directly.
If it is a triangle question, label opposite, adjacent and hypotenuse before choosing sine, cosine or tangent.

Self review

Check yourself

Can you write down $sin⁡(30∘)\sin(30^\circ)$ , $cos⁡(60∘)\cos(60^\circ)$ and $tan⁡(45∘)\tan(45^\circ)$ without a calculator?
Which exact trig values involve $2\sqrt{2}$ ?
In a right-angled triangle, how do you decide which side is opposite?

Exact trig values

What you'll learn

What “exact” means in trigonometry.
The exact values of sine, cosine and tangent for 0°, 30°, 45°, 60° and 90°.
How to use special triangles to remember the values.
How to use exact trig values to find missing lengths in right-angled triangles.

What does “exact” mean?

In this topic, you are not estimating with a calculator. You are giving the precise value.

Definition

Exact value

An exact value is an answer written without rounding. It might be a whole number, a fraction, or a surd, which is a square root that does not simplify to a whole number, such as $2\sqrt{2}$ .

For example, 0.707 is only an approximation. The exact value of $sin⁡(45∘)\sin(45^\circ)$ is $22\frac{\sqrt{2}}{2}$ .

Example

Exact, not rounded

Suppose you are asked for $sin⁡(45∘)\sin(45^\circ)$ .
A calculator gives a decimal starting 0.7071..., but this is rounded.
The exact GCSE answer is:
$sin⁡(45∘)=22\sin(45^\circ)=\frac{\sqrt{2}}{2}$

Common Mistake

Using decimals too early

If the question asks for an exact value, do not write a rounded decimal like 0.866. Use the fraction or surd form instead.

Right-angled triangle basics

Exact trig values come from special right-angled triangles.

Definition

The three side names

The hypotenuse is the longest side of a right-angled triangle. It is opposite the right angle.
The opposite side is opposite the angle you are using.
The adjacent side is next to the angle you are using, but it is not the hypotenuse.

The hypotenuse is opposite the right angle, while opposite and adjacent are named relative to the chosen angle θ.

We often call the angle we are using $θ\theta$ , pronounced “theta”.

Key Idea

SOHCAHTOA

Example

Choosing the correct trig ratio

Imagine a right-angled triangle where the side opposite $θ\theta$ is 6 cm and the hypotenuse is 12 cm.

For this triangle, the given sides are opposite θ and hypotenuse, so the sine ratio is used.

Opposite and hypotenuse means use sine:
$sin⁡θ=oppositehypotenuse\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$
Substitute the side lengths:
$sin⁡θ=612\sin\theta=\frac{6}{12}$
Simplify:
$sin⁡θ=12\sin\theta=\frac{1}{2}$

The 45° special triangle

An isosceles triangle has two equal sides. If you make an isosceles right-angled triangle, the two smaller angles are both 45°.

The useful side lengths are:

The 45° special triangle has equal shorter sides 1 and 1, with hypotenuse √2.

two shorter sides: 1 and 1
hypotenuse: $2\sqrt{2}$

So for a 45° angle, the opposite and adjacent sides are the same.

Example

Finding an exact value at 45°

Use the 45-45-90 triangle with shorter sides 1 and 1, and hypotenuse $2\sqrt{2}$ .

Using either 45° angle, the opposite side is 1 and the hypotenuse is √2.

For $sin⁡(45∘)\sin(45^\circ)$ , use opposite over hypotenuse:
$sin⁡(45∘)=12\sin(45^\circ)=\frac{1}{\sqrt{2}}$
Remove the square root from the denominator:
$12=22\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$
Therefore:
$sin⁡(45∘)=22\sin(45^\circ)=\frac{\sqrt{2}}{2}$

Tip

45° shortcut

For 45°, sine and cosine are equal: $sin⁡(45∘)=cos⁡(45∘)=22\sin(45^\circ)=\cos(45^\circ)=\frac{\sqrt{2}}{2}$ .

The 30° and 60° special triangle

An equilateral triangle has all three sides equal and all three angles 60°. If you cut one in half, you get a 30-60-90 right-angled triangle.

Cutting an equilateral triangle in half creates a 30°–60°–90° right-angled triangle.

The useful side lengths are:

shortest side: 1
middle side: $3\sqrt{3}$
hypotenuse: 2

For the 30° angle, the opposite side is the shortest side.
For the 60° angle, the opposite side is the $3\sqrt{3}$ side.

In the 30°–60°–90° triangle, the side opposite 60° is √3 and the side opposite 30° is 1.

Example

Finding an exact value at 60°

Use the 30-60-90 triangle with side lengths 1, $3\sqrt{3}$ and 2.
For a 60° angle, the opposite side is $3\sqrt{3}$ and the adjacent side is 1.
Tangent uses opposite over adjacent:
$tan⁡(60∘)=31\tan(60^\circ)=\frac{\sqrt{3}}{1}$
So $tan⁡(60∘)=3\tan(60^\circ)=\sqrt{3}$ .

Common Mistake

Swapping 30° and 60°

For sine, 30° has the smaller value $12\frac{1}{2}$ and 60° has the larger value $32\frac{\sqrt{3}}{2}$ . Cosine is the other way round.

The exact values to know

Key Idea

Memorise these values

0°: $sin⁡(0∘)=0\sin(0^\circ)=0$ , $cos⁡(0∘)=1\cos(0^\circ)=1$ , $tan⁡(0∘)=0\tan(0^\circ)=0$
30°: $sin⁡(30∘)=12\sin(30^\circ)=\frac{1}{2}$ , $cos⁡(30∘)=32\cos(30^\circ)=\frac{\sqrt{3}}{2}$ , $tan⁡(30∘)=33\tan(30^\circ)=\frac{\sqrt{3}}{3}$
45°: $sin⁡(45∘)=22\sin(45^\circ)=\frac{\sqrt{2}}{2}$ , $cos⁡(45∘)=22\cos(45^\circ)=\frac{\sqrt{2}}{2}$ , $tan⁡(45∘)=1\tan(45^\circ)=1$
60°: $sin⁡(60∘)=32\sin(60^\circ)=\frac{\sqrt{3}}{2}$ , $cos⁡(60∘)=12\cos(60^\circ)=\frac{1}{2}$ , $tan⁡(60∘)=3\tan(60^\circ)=\sqrt{3}$
90°: $sin⁡(90∘)=1\sin(90^\circ)=1$ , $cos⁡(90∘)=0\cos(90^\circ)=0$

Common Mistake

Why tan 90° is missing

$tan⁡(90∘)\tan(90^\circ)$ is not defined because tangent involves dividing by the adjacent side, and at 90° this would mean dividing by zero.

Example

Writing down exact values

For $cos⁡(90∘)\cos(90^\circ)$ , look at the 90° values: $cos⁡(90∘)=0\cos(90^\circ)=0$ .
For $tan⁡(30∘)\tan(30^\circ)$ , look at the 30° values: $tan⁡(30∘)=33\tan(30^\circ)=\frac{\sqrt{3}}{3}$ .
For $sin⁡(60∘)\sin(60^\circ)$ , look at the 60° values: $sin⁡(60∘)=32\sin(60^\circ)=\frac{\sqrt{3}}{2}$ .

Tip

Sine pattern

Using exact trig values to find lengths

You may also be given a right-angled triangle and asked to calculate a missing side. The method is the same as normal trigonometry, but you substitute an exact value instead of a calculator decimal.

Example

Finding a hypotenuse using 30°

A right-angled triangle has a 30° angle. The side opposite this angle is 18 cm. Let the hypotenuse be $h$ .

The side opposite the 30° angle is given as 18 cm and the hypotenuse is h.

Opposite and hypotenuse means use sine:
$sin⁡(30∘)=18h\sin(30^\circ)=\frac{18}{h}$
Substitute the exact value $sin⁡(30∘)=12\sin(30^\circ)=\frac{1}{2}$ :
$12=18h\frac{1}{2}=\frac{18}{h}$
Solve for $h$ :
$h = 36$
The hypotenuse is 36 cm.

Example

Finding an adjacent side using 60°

A right-angled triangle has a 60° angle. The side opposite this angle is 18 cm. Let the adjacent side be $a$ .

The side opposite the 60° angle is 18 cm and the adjacent side to find is a.

Opposite and adjacent means use tangent:
$tan⁡(60∘)=18a\tan(60^\circ)=\frac{18}{a}$
Substitute the exact value $tan⁡(60∘)=3\tan(60^\circ)=\sqrt{3}$ :
$3=18a\sqrt{3}=\frac{18}{a}$
Rearrange and simplify:
$a=183=1833=63a=\frac{18}{\sqrt{3}}=\frac{18\sqrt{3}}{3}=6\sqrt{3}$
The adjacent side is $636\sqrt{3}$ cm.

Exam technique

In the exam

First identify the angle: is it 0°, 30°, 45°, 60° or 90°?
If it is a “write down” question, use the memorised exact value directly.
If it is a triangle question, label opposite, adjacent and hypotenuse before choosing sine, cosine or tangent.

Self review

Check yourself

Can you write down $sin⁡(30∘)\sin(30^\circ)$ , $cos⁡(60∘)\cos(60^\circ)$ and $tan⁡(45∘)\tan(45^\circ)$ without a calculator?
Which exact trig values involve $2\sqrt{2}$ ?
In a right-angled triangle, how do you decide which side is opposite?

Exact trig values

What you'll learn

What does “exact” mean?

Right-angled triangle basics

The 45° special triangle

The 30° and 60° special triangle

The exact values to know

Using exact trig values to find lengths

Recap questions

Exact trig values

What you'll learn

What does “exact” mean?

Right-angled triangle basics

The 45° special triangle

The 30° and 60° special triangle

The exact values to know

Using exact trig values to find lengths

Recap questions

Exact trig values

What you'll learn

What does “exact” mean?

Right-angled triangle basics

The 45° special triangle

The 30° and 60° special triangle

The exact values to know

Using exact trig values to find lengths

Recap questions

Exact trig values Revision Guide

Exact trig values

What you'll learn

What does “exact” mean?

Right-angled triangle basics

The 45° special triangle

The 30° and 60° special triangle

The exact values to know

Using exact trig values to find lengths

Recap questions

Exact trig values Revision Guide