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 $0^\circ$ , $30^\circ$ , $45^\circ$ , $60^\circ$ , and $90^\circ$ .

Visualization of Special Angles on the Unit Circle

Move the slider to angles

0^\circ

30^\circ

45^\circ

60^\circ

, or

90^\circ

to see the positions of special angles.

Sin (45°) = 0.71Cos (45°) = 0.71Tan (45°) = 1.00

45°0.79 Radian

Exact Values of Trigonometric Ratios for Special Angles

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

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

Origin of Exact Values

The exact values for special angles are derived from:

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