# Functions of a Right Triangle

From the right triangle shown below,

the trigonometric functions of angle θ are defined as follows:

$\sin \theta = \dfrac{opposite\,\,side}{hypotenuse} = \dfrac{a}{c}$

$\cos \theta = \dfrac{adjacent\,\,side}{hypotenuse} = \dfrac{b}{c}$

$\tan \theta = \dfrac{opposite\,\,side}{adjacent\,\,side} = \dfrac{a}{b}$

$\csc \theta = \dfrac{hypotenuse}{opposite\,\,side} = \dfrac{c}{a}$

$\sec \theta = \dfrac{hypotenuse}{adjacent\,\,side} = \dfrac{c}{b}$

$\cot \theta = \dfrac{adjacent\,\,side}{opposite\,\,side} = \dfrac{b}{a}$

The above relationships can be written into acronym soh-cah-toa-cho-sha-cao.

**soh**=**s**ine of theta is equal to**o**pposite side over the**h**ypotenuse.**cah**=**c**osine of theta is equal to**a**djacent side over the**h**ypotenuse.**toa**=**t**angent of theta is equal to**o**pposite side over the**a**djacent side.**cho**=**c**osecant of theta is equal to**h**ypotenuse over the**o**pposite side.**sha**=**s**ecant of theta is equal to**h**ypotenuse over the**a**djacent side.**cao**=**c**otangent of theta is equal to**a**djacent side over the**o**pposite side.

See how these relationships were used to derive the Pythagorean identities.

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