Special Angles in Trigonometric Comparisons

Exact Trigonometric Values from Triangles and the Unit Circle

Special angles are angles whose trigonometric values can be written exactly without a calculator. Their values follow from two familiar right triangles and from coordinates on the unit circle.

The most common special angles are $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, and $90^\circ$.

Where the Exact Values Come From

The exact values for $30^\circ$, $45^\circ$, and $60^\circ$ follow from two triangles whose side lengths are known exactly. One triangle comes from half of a square and the other from half of an equilateral triangle, so both sets of ratios can be read off without a calculator.

The Forty Five Degree Triangle

Cutting a square along a diagonal produces an isosceles right triangle. If both legs have length $1$, the Pythagorean theorem gives a hypotenuse of $\sqrt{2}$:

Triangles with Thirty and Sixty Degree Angles

Bisecting an equilateral triangle of side length $2$ produces two right triangles. Their side lengths, opposite $30^\circ$, opposite $60^\circ$, and along the hypotenuse, are $1$, $\sqrt{3}$, and $2$:

The values at $0^\circ$ and $90^\circ$ come from the unit-circle points $(1,0)$ and $(0,1)$. Because tangent is $\sin \theta/\cos \theta$, $\tan 90^\circ$ is undefined: its denominator is zero.

Exact Value Table

The derivations above produce this table: