Exact trigonometric values

Lesson overview

Geometry — trigonometry

The exact trigonometric values for special angles come from two key triangles: the **isosceles right-angled triangle** (for $45°$ ) and the **equilateral triangle split in half** (for $30°$ and $60°$ ). $\sin 30° = \dfrac{1}{2}, \quad \sin 45° = \dfrac{\sqrt{2}}{2}, \quad \sin 60° = \dfrac{\sqrt{3}}{2}$ $\cos 30° = \dfrac{\sqrt{3}}{2}, \quad \cos 45° = \dfrac{\sqrt{2}}{2}, \quad \cos 60° = \dfrac{1}{2}$ $\tan 30° = \dfrac{1}{\sqrt{3}}, \quad \tan 45° = 1, \quad \tan 60° = \sqrt{3}$ At $0°$ and $90°$ : $\sin 0° = 0$ , $\sin 90° = 1$ , $\cos 0° = 1$ , $\cos 90° = 0$ , $\tan 0° = 0$ , $\tan 90°$ is undefined.

The special triangles give us exact values

An equilateral triangle of side $2$ split in half gives a right triangle with sides $1$ , $\sqrt{3}$ , and $2$ .

Half of an equilateral triangle of side 2

\sin 30°

$\dfrac{1}{2}$

opposite/hypotenuse

\cos 30°

$\dfrac{\sqrt{3}}{2}$

adjacent/hypotenuse

\tan 30°

$\dfrac{1}{\sqrt{3}}$

opposite/adjacent

The ratios of sides in these special triangles give exact trig values.

The $45°$ isosceles right triangle

A square of side $1$ cut along the diagonal gives a right triangle with two equal sides of $1$ and hypotenuse $\sqrt{2}$ .

When the two shorter sides are equal, $\sin = \cos$ .

Memorising the pattern

There is a pattern: $\sin$ values go $\dfrac{\sqrt{0}}{2}, \dfrac{\sqrt{1}}{2}, \dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{3}}{2}, \dfrac{\sqrt{4}}{2}$ for $0°, 30°, 45°, 60°, 90°$ .

$\theta$	$0°$	$30°$	$45°$	$60°$	$90°$
$\sin \theta$	$0$	$\dfrac{1}{2}$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{3}}{2}$	$1$
$\cos \theta$	$1$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{1}{2}$	$0$
$\tan \theta$	$0$	$\dfrac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	undef.

$\cos$ values go in reverse order of $\sin$ . $\tan = \dfrac{\sin}{\cos}$ .

Worked example

Without a calculator, find the exact value of $2\sin 60° + \cos 30°$ .

$\sin 60° = \dfrac{\sqrt{3}}{2}$ and $\cos 30° = \dfrac{\sqrt{3}}{2}$ .
$2\sin 60° = 2 \times \dfrac{\sqrt{3}}{2} = \sqrt{3}$ .
So the expression $= \sqrt{3} + \dfrac{\sqrt{3}}{2} = \dfrac{2\sqrt{3} + \sqrt{3}}{2}$ .
$= \dfrac{3\sqrt{3}}{2}$ .

Try it

Recall the exact values from the table, then substitute carefully.

Question 1

What is the exact value of $\sin 30°$ ?

Question 2

What is the exact value of $\cos 60°$ ?

Question 3

What is the exact value of $\tan 45°$ ?

Question 4

Without a calculator, find $\sin^2 30° + \cos^2 30°$ .

Question 5

Which trigonometric function is undefined at $90°$ ?

Common mistakes

Watch for these when working through the lesson.

Confusing $\sin 30° = \frac{1}{2}$ with $\sin 60° = \frac{\sqrt{3}}{2}$ .
Forgetting that $\tan 90°$ is undefined (division by zero).
Writing $\dfrac{\sqrt{2}}{3}$ instead of $\dfrac{\sqrt{3}}{2}$ — the denominator is always $2$ for $\sin$ and $\cos$ .

Exact trigonometric values

Lesson overview

Worked example

Try it

Common mistakes

Related topics

Independent practice will plug in here