Trigonometry/Soh-Cah-Toa

The Mnemonic

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$Soh-Cah-Toa$

$\begin{array}{ccc}\sin (A)& =& \frac{\text{opposite side}}{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{u}\mathrm{s}\mathrm{e}}\\ \\ \cos (A)& =& \frac{\text{adjacent side}}{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{u}\mathrm{s}\mathrm{e}}\\ \\ \tan (A)& =& \frac{\text{opposite side}}{\text{adjacent side}}\end{array}$

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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:

$\sin (A)=\frac{a}{c}$ or opposite side over hypotenuse

$\cos (A)=\frac{b}{c}$ or adjacent side over hypotenuse

$\tan (A)=\frac{a}{b}$ 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.

• Basic Trigonometry Soh-Cah-Toa (sin and cosine)
• Basic Trigonometry II Soh-Cah-Toa (starting with tan)

Exercises: (Draw a diagram!)

• A right triangle has side a = 3, b = 4, and c = 5.
• Calculate the following:

$\sin (A),\cos (A),\tan (A)$

• A different right triangle has side $c=6$ and $\sin (A)=0.5$ .
• Calculate side a.

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