Trigonometry basics: SOH CAH TOA and exact values

The three trig ratios, the two special triangles that give exact values, and how degrees relate to radians.

阅读约 1 分钟更新于 2026年10月18日作者：Sumzo team审核：Sumzo team

In a right triangle, label the sides relative to an angle $\theta$ : the hypotenuse (longest side), the opposite side (across from $\theta$ ) and the adjacent side (next to $\theta$ ).

\sin\theta = \frac{\text{opp}}{\text{hyp}},\quad \cos\theta = \frac{\text{adj}}{\text{hyp}},\quad \tan\theta = \frac{\text{opp}}{\text{adj}}

SOH CAH TOA

Exact values from special triangles

Half of an equilateral triangle with side 2 has sides $1, \sqrt{3}, 2$ and angles 30°, 60°, 90°. Half of a unit square has sides $1, 1, \sqrt{2}$ and angles 45°, 45°, 90°. Reading the ratios off these triangles gives exact values.

例题

Exact values at 30°

Find the exact values of $\sin 30^\circ$ , $\cos 30^\circ$ and $\tan 30^\circ$ .

显示解题过程

In the 30-60-90 triangle, the side opposite 30° is 1, the adjacent side is √3 and the hypotenuse is 2. $\sin 30^\circ = \tfrac{1}{2},\quad \cos 30^\circ = \tfrac{\sqrt{3}}{2}$
Divide sine by cosine and rationalise. $\tan 30^\circ = \tfrac{1}{\sqrt{3}} = \tfrac{\sqrt{3}}{3}$

答案： $\tfrac{1}{2}$ , $\tfrac{\sqrt{3}}{2}$ and $\tfrac{\sqrt{3}}{3}$

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Degrees and radians

A full turn is 360° or $2\pi$ radians, so $180^\circ = \pi$ . Multiply by $\frac{\pi}{180}$ to convert degrees to radians.

例题

Convert 30° to radians

Write 30° in radians.