Ratios of Standard Angles (0°, 30°, 45°, 60°, 90°)

Five angles — $0°, 30°, 45°, 60°, 90°$ — appear so often in numerical work that their trigonometric ratios are worth memorising as a single reference table rather than recomputed from a triangle every time.

The $45°$ angle comes from an isosceles right-angled triangle (a right triangle with both acute angles equal to $45°$). If both legs are taken as length 1, the hypotenuse is $\sqrt{1^2+1^2} = \sqrt2$ by the Pythagorean theorem, giving $\sin45° = \cos45° = \dfrac{1}{\sqrt2}$ and $\tan45° = 1$.

The $30°$ and $60°$ angles come from an equilateral triangle of side 2, split into two right triangles by an altitude. The altitude bisects the base and the apex angle, so each half-triangle has angles $30°, 60°, 90°$, a base of length 1, an altitude of $\sqrt{2^2-1^2} = \sqrt3$, and a hypotenuse of 2 — giving $\sin30° = \tfrac12$, $\cos30° = \tfrac{\sqrt3}{2}$, $\sin60° = \tfrac{\sqrt3}{2}$, $\cos60° = \tfrac12$.

The $0°$ and $90°$ angles are the limiting cases as the angle in a right triangle shrinks to nothing or grows to fill the whole right angle; their ratios are read off directly ($\sin0°=0,\ \cos0°=1$ and $\sin90°=1,\ \cos90°=0$), and $\tan90°$, $\cot0°$ are undefined because they would require dividing by zero.