# Trig values of 45 degrees

## Statement

The exact value of the trigonometry functions of $45 \degree$ are:

$$\sin(45 \degree) = \tfrac{1}{2}\sqrt{2}$$

$$\cos(45 \degree) = \tfrac{1}{2}\sqrt{2}$$

$$\tan(45 \degree) = 1$$

## Proof

Construct triangle $ABC$ with $\angle A = \angle C = 45 \degree$ and $\angle B = 90 \degree$, like the image below.

Note that this triangle is an isosceles triangle with $AB = BC$. Now call those sides $m$.

Calculate $AC$ using the Pythagorean theorem:

$$AC = \sqrt{AB^2 + BC^2} = \sqrt{m^2 + m^2} = \sqrt{2m^2} = m\sqrt{2}$$

### Sine

From the triangle, note that $\sin(\angle A) = \dfrac{BC}{AC}$, so:

$$\sin(45 \degree) = \frac{m}{m\sqrt{2}} = \frac{1}{\sqrt{2}} = \tfrac{1}{2}\sqrt{2}$$

### Cosine

From the triangle, note that $\cos(\angle A) = \dfrac{AB}{AC}$, so:

$$\cos(45 \degree) = \frac{m}{m\sqrt{2}} = \frac{1}{\sqrt{2}} = \tfrac{1}{2}\sqrt{2}$$

### Tangent

From the triangle, note that $\tan(\angle A) = \dfrac{BC}{AB}$, so:

$$\tan(45 \degree) = \frac{m}{m} = 1$$

## Proofs building upon this proof

### Exact values of the trig functions

This table shows all exact values of sine, cosine and tangent.

### Trig values of 15 degrees

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