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.

1. soh = sine of theta is equal to opposite side over the hypotenuse.
2. cah = cosine of theta is equal to adjacent side over the hypotenuse.
3. toa = tangent of theta is equal to opposite side over the adjacent side.
4. cho = cosecant of theta is equal to hypotenuse over the opposite side.
5. sha = secant of theta is equal to hypotenuse over the adjacent side.
6. cao = cotangent of theta is equal to adjacent side over the opposite side.

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