# Tangent is sine divided by cosine

## 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:

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:

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

## Proofs building upon this proof

### Derivative of tan(x)

This proof shows what the derivative of tan(x) is.

### The limit of sin(x) over x

This proof shows that the limit as x goes to zero of sin(x) over x converges to one.

### Trig values of 15 degrees

This proof shows the exact value of sin(15), cos(15) and tan(15)

### Trig values of 3 degrees

This proof shows the exact value of sin(3), cos(3) and tan(3).

### Trig values of 36 degrees

This proof shows the exact value of sin(36), cos(36) and tan(36)

### Trig values of 54 degrees

This proof shows the exact value of sin(54), cos(54) and tan(54)