Special Angles in Trigonometric Comparisons

What are Special Angles?

Special angles in trigonometric ratios are angles whose ratio values can be determined exactly (precise values) without using a calculator. These values are very important to remember and will be frequently used in various mathematical and physics calculations.

The most common special angles are $0°$ , $30°$ , $45°$ , $60°$ , and $90°$ .

Visualization of Special Angles on the Unit Circle

Move the slider to angles

0°

30°

45°

60°

, or

90°

to see the positions of special angles.

45°0.79 Radian

0°360°

Exact Values of Trigonometric Ratios for Special Angles

Below is a table of exact values for trigonometric ratios at special angles:

Ratio	$0°$	$30°$	$45°$	$60°$	$90°$
$\sin \theta$	$0$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$
$\cos \theta$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$
$\tan \theta$	$0$	$\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$	$1$	$\sqrt{3}$	Undefined

Origin of Exact Values

The exact values for special angles are derived from:

Angles $30°$ and $60°$ : Obtained from a right triangle with side ratio $1 : 2 : \sqrt{3}$ .
Angle $45°$ : Obtained from an isosceles right triangle with side ratio $1 : 1 : \sqrt{2}$ .

Command Palette

Special Angles in Trigonometric Comparisons

What are Special Angles?

Exact Values of Trigonometric Ratios for Special Angles

Origin of Exact Values