# Angle sum/difference formula for sine and cosine

## Statement

When adding or subtracting angles in sine or cosine, these formulae apply for $\alpha, \beta \in \R$:

$$\sin(\alpha + \beta) = \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta)$$

$$\sin(\alpha - \beta) = \sin(\alpha) \cos(\beta) - \cos(\alpha) \sin(\beta)$$

$$\cos(\alpha + \beta) = \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta)$$

$$\cos(\alpha - \beta) = \cos(\alpha) \cos(\beta) + \sin(\alpha) \sin(\beta)$$

## Proof

Use Euler's formula:

$$e^{\theta i} = \cos(\theta) + i\sin(\theta)$$

When substituting $\alpha + \beta$ for $\theta$, you get:

$$e^{(\alpha + \beta)i} = \cos(\alpha + \beta) + i\sin(\alpha + \beta)$$

Now, take $e^{(\alpha + \beta)i}$ again and expand the brackets. Use that $i^2 = -1$.

\begin{aligned}e^{(\alpha + \beta)i} &= e^{\alpha i + \beta i} \newline&= e^{\alpha i} \cdot e^{\beta i} \newline&= \bigg(\cos(\alpha) + i\sin(\alpha)\bigg) \cdot \bigg(\cos(\beta) + i\sin(\beta)\bigg) \newline&= \cos(\alpha) \cos(\beta) + i \cos(\alpha) \sin(\beta) + i \sin(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta) \newline&= \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta) + i \bigg(\cos(\alpha) \sin(\beta) + \sin(\alpha) \cos(\beta)\bigg)\end{aligned}

Since both expressions are equal to $e^{(\alpha + \beta)i}$, set them equal to eachother:

$$\cos(\alpha + \beta) + i \sin(\alpha + \beta) = \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta) + i \bigg(\cos(\alpha) \sin(\beta) + \sin(\alpha) \cos(\beta)\bigg)$$

Since sine and cosine give a real value for a real input, the real parts and imaginary parts must match, giving:

$$\cos(\alpha + \beta) = \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta)$$

$$\sin(\alpha + \beta) = \cos(\alpha) \sin(\beta) + \sin(\alpha) \cos(\beta)$$

When substituting $- \beta$ for $\beta$, use that $\sin(-\theta) = -\sin(\theta)$ and $\cos(-\theta) = \cos(\theta)$:

$$\cos(\alpha - \beta) = \cos(\alpha) \cos(- \beta) - \sin(\alpha) \sin(-\beta) = \cos(\alpha) \cos(\beta) + \sin(\alpha) \sin(\beta)$$

$$\sin(\alpha - \beta) = \cos(\alpha) \sin(- \beta) + \sin(\alpha) \cos(- \beta) = \sin(\alpha) \cos(\beta) - \cos(\alpha) \sin(\beta)$$

## Proofs building upon this proof

### Derivative of cos(x)

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

### Derivative of sin(x)

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

### 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)