# What are the six trigonometric functions of a right triangle?

Oct 11, 2014

Given a right triangle, select a vertex where the hypotenuse and one of the legs meet.

The angle $\theta$ created by this two sides will be used as a reference point

The side that formed the angle $\theta$ together with the hypotenuse will be referred to as $a \mathrm{dj} a c e n t$ (side adjacent to the angle). The other side will be referred to as $o p p o s i t e$ (side opposite the angle)

The ratio between the $o p p o s i t e$ and the $\text{hypotenuse}$ is called "sine" (sin). The inverse of this ratio is called "cosecant" (csc)

$\sin \theta = \text{opposite" / "hypotenuse}$

$\csc \theta = \text{hypotenuse" / "opposite} = \frac{1}{\sin} \theta$

The ratio between the $a \mathrm{dj} a c e n t$ and the $\text{hypotenuse}$ is called "cosine". The inverse of this ratio is called "secant"

$\cos \theta = \text{adjacent" / "hypotenuse}$

$\sec \theta = \text{hypotenuse" / "adjacent} = \frac{1}{\cos} \theta$

The ratio between the $o p p o s i t e$ and the $a \mathrm{dj} a c e n t$ is called
$\text{tangent}$. The inverse of this ratio is called $\text{cotangent}$

$\tan \theta = \text{opposite" / "adjacent}$

$\cot \theta = \text{adjacent" / "opposite} = \frac{1}{\tan} \theta$

For example, in a 30-60-90 triangle

$\sin 30 = \frac{1}{2}$
$\cos 30 = {3}^{\frac{1}{2}} / 2$
$\tan 30 = \frac{1}{3} ^ \left(\frac{1}{2}\right)$
$\csc 30 = 2$
$\sec 30 = \frac{2}{3} ^ \left(\frac{1}{2}\right)$
$\cot 30 = {3}^{\frac{1}{2}}$

$\sin 60 = {3}^{\frac{1}{2}} / 2$
$\cos 60 = \frac{1}{2}$
$\tan 60 = {3}^{\frac{1}{2}}$
$\csc 60 = \frac{2}{3} ^ \left(\frac{1}{2}\right)$
$\sec 60 = 2$
$\cot 60 = \frac{1}{3} ^ \left(\frac{1}{2}\right)$