Trig 30/45/60 — Trig at 30° / 45° / 60° · Principia

Depends on: 30-60-90 triangle side ratios (H.04 derived / D.09), definition of sin / cos / tan (Definition of sin / cos / tan), Pythagorean theorem (Pythagorean theorem), base angles of an isosceles triangle are equal (Base angles of an isoceles triangle are equal).

Statement

The table below lists the 9 special values that recur most often in elementary geometry / trigonometry:

\begin{array}{c|ccc} & 30^\circ & 45^\circ & 60^\circ \\\hline \sin & \tfrac{1}{2} & \tfrac{\sqrt{2}}{2} & \tfrac{\sqrt{3}}{2} \\ \cos & \tfrac{\sqrt{3}}{2} & \tfrac{\sqrt{2}}{2} & \tfrac{1}{2} \\ \tan & \tfrac{\sqrt{3}}{3} & 1 & \sqrt{3} \end{array}

All of them are read off two reference right triangles: the 30-60-90 triangle gives the $30^\circ$ and $60^\circ$ columns, and the isosceles right triangle (45-45-90) gives the $45^\circ$ column.

$Table of special trig values at 30°, 45°, 60° together with the two reference right triangles (1, \sqrt{3}, 2) and (1, 1, \sqrt{2}).$

Special trigonometric values at 30°, 45°, 60°

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