# How are the 6 basic trigonometric functions related to right triangles?

Nov 26, 2014

The six basic trigonometric functions are:
1. Sine, $\sin \theta$
2. Cosine, $\cos \theta$
3. Tangent, $\tan \theta$
4. Cotangent, $\cot \theta$
5. Secant, $\sec \theta$
6. Cosecant, $\csc \theta$

Take the following triangle for example:

Let the angle marked at A be $\theta$.
The longest side of the triangle is the hypotenuse, the side next to the angle is the adjacent and the side opposite to it is the opposite.

Then,

$\sin \theta = \frac{o p p}{h y p} = \frac{a}{h}$

$\cos \theta = \frac{a \mathrm{dj}}{h y p} = \frac{b}{h}$

$\tan \theta = \frac{o p p}{a \mathrm{dj}} = \frac{a}{b} = \frac{\sin \theta}{\cos \theta}$

$\cot \theta = \frac{a \mathrm{dj}}{o p p} = \frac{b}{a} = \frac{\cos \theta}{\sin \theta} = \frac{1}{\tan} \theta$

$\sec \theta = \frac{h y p}{a \mathrm{dj}} = \frac{h}{b} = \frac{1}{\cos} \theta$

$\csc \theta = \frac{h y p}{o p p} = \frac{h}{a} = \frac{1}{\sin} \theta$

Now if we the triangle were to have sides 3, 4 and 5 units long such that
$a = 4 , b = 3 , h = 5$
The six basic trigonometric functions would be:

$\sin \theta = \frac{a}{h} = \frac{4}{5}$

$\cos \theta = \frac{b}{h} = \frac{3}{5}$

$\tan \theta = \frac{a}{b} = \frac{4}{3}$

$\cot \theta = \frac{b}{a} = \frac{3}{4}$

$\sec \theta = \frac{h}{b} = \frac{5}{3}$

$\csc \theta = \frac{h}{a} = \frac{5}{4}$