# Trigonometry/Soh-Cah-Toa

## The Mnemonic

$Soh-Cah-Toa\,$

$\sin A = \frac{opposite\,side}{hypotenuse}\,$

$\cos A = \frac{adjacent\,side}{hypotenuse}\,$

$\tan A = \frac{opposite\,side}{adjacent\,side}\,$

Soh-Cah-Toa is a mnemonic for remembering how to use sines cosines and tangents to compute lengths in right angle triangles.

# Trigonometric definitions

This triangle has sides a and b. The angle between them, C, is a right angle. The third side, c, is the hypotenuse. Side a is opposite angle A, and side b is adjacent to angle A.

Applying the definitions of the functions, we have:

$\displaystyle\sin(A) = a / c$ or opposite side over hypotenuse
$\displaystyle\cos(A) = b / c$ or adjacent side over hypotenuse
$\displaystyle\tan(A) = a / b$ or opposite side over adjacent side

This can be memorized using the mnemonic 'SOH-CAH-TOA' (sin = opposite over hypothenuse, cosine = adjacent over hypotenuse, tangent = opposite over adjacent).

 Exercise: Find sin cos and tan A right triangle has side a = 3, b = 4, and c = 5. Calculate the following: $\displaystyle \sin(A), \cos(A), \tan(A)$
 Exercise: Find a side A different right triangle has side $\displaystyle c = 6$ and $\displaystyle\sin(A) = 0.5$ . Calculate side a.