Tangent is sine divided by cosine

Geometry

Statement

Sine divided by cosine is tangent.

$$ \tan(x) = \frac{\sin(x)}{\cos(x)} $$

Proof

Define the right triangle $ abc $ with one angle being $ \theta $, like the image below:

A right triangle with sides a, b and c

Note the following:

$$ \sin(\theta) = \frac{b}{c} $$

$$ \cos(\theta) = \frac{a}{c} $$

Now divide $ \sin(\theta) $ by $ \cos(\theta) $ and simplify.

$$ \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{b}{c}}{\frac{a}{c}} = \frac{bc}{ac} = \frac{b}{a} $$

Note from the image above that too

$$ \tan(\theta) = \frac{b}{a} $$

And thus:

$$ \tan(x) = \frac{\sin(x)}{\cos(x)} $$