Trigonometry/Soh-Cah-Toa

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The Mnemonic

${\displaystyle Soh-Cah-Toa}$

${\displaystyle \displaystyle {\begin{array}{ccc}\sin(A)&=&\displaystyle {\frac {\text{opposite side}}{\rm {hypotenuse}}}\\\\\cos(A)&=&\displaystyle {\frac {\text{adjacent side}}{\rm {hypotenuse}}}\\\\\tan(A)&=&\displaystyle {\frac {\text{opposite side}}{\text{adjacent side}}}\end{array}}}$

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:

${\displaystyle \sin(A)={\frac {a}{c}}}$ or opposite side over hypotenuse
${\displaystyle \cos(A)={\frac {b}{c}}}$ or adjacent side over hypotenuse
${\displaystyle \tan(A)={\frac {a}{b}}}$ 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).

Video Link

These links from the Khan Academy are useful for this topic.

Exercises: (Draw a diagram!)

 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.