Formulas of trigonometric functions

sum and difference

$$\begin{align*} \sin(\alpha + \beta) &= \sin\alpha \cos\beta + \cos\alpha \sin\beta \\ \sin(\alpha - \beta) &= \sin\alpha \cos\beta - \cos\alpha \sin\beta \\ \end{align*}$$

$$\begin{align*} \cos(\alpha + \beta) &= \cos\alpha \cos\beta - \sin\alpha \sin\beta \\ \cos(\alpha - \beta) &= \cos\alpha \cos\beta + \sin\alpha \sin\beta \\ \end{align*}$$

double angle

$$\begin{align*} \sin 2 \alpha &= 2 \sin \alpha \cos \alpha \end{align*}$$

$$\begin{align*} \cos 2 \alpha &= \cos^2 \alpha - \sin^2 \alpha \\ &= 2 \cos^2 \alpha - 1 \\ &= 1 - 2 \sin^2 \alpha \\ \end{align*}$$

half angle

$$\begin{align*} \sin^2 \alpha = \frac{1 - \cos 2 \alpha}{2} \end{align*}$$

$$\begin{align*} \cos^2 \alpha = \frac{1 + \cos 2 \alpha}{2} \end{align*}$$

product to sum

$$\begin{align*} \sin\alpha \cos\beta &= \frac{1}{2} \left( \sin(\alpha + \beta) + \sin(\alpha - \beta) \right) \\ \cos\alpha \sin\beta &= \frac{1}{2} \left( \sin(\alpha + \beta) - \sin(\alpha - \beta) \right) \\ \cos\alpha \cos\beta &= \frac{1}{2} \left( \cos(\alpha + \beta) + \cos(\alpha - \beta) \right) \\ \sin\alpha \sin\beta &= \frac{1}{2} \left( \cos(\alpha - \beta) - \cos(\alpha + \beta) \right) \\ \end{align*}$$

$$\begin{align*} \end{align*}$$

sum to product

$$\begin{align*} \sin\alpha + \sin\beta &= 2 \sin\frac{\alpha + \beta}{2} \cos\frac{\alpha - \beta}{2} \\ \sin\alpha - \sin\beta &= 2 \cos\frac{\alpha + \beta}{2} \sin\frac{\alpha - \beta}{2} \\ \cos\alpha + \cos\beta &= 2 \cos\frac{\alpha + \beta}{2} \cos\frac{\alpha - \beta}{2} \\ \cos\alpha - \cos\beta &= -2 \sin\frac{\alpha + \beta}{2} \sin\frac{\alpha - \beta}{2} \\ \end{align*}$$

integral

$$\begin{align*} \int_{-\pi}^\pi \cos(mx)dx &= 0 \\ \int_{-\pi}^\pi \sin(mx)dx &= 0 \\ \int_{-\pi}^\pi \sin(mx) \cos(nx) dx &= 0 \\ \int_{-\pi}^\pi \cos(mx) \cos(nx) dx &= \begin{cases} \pi,\, (m = n) \\ 0,\, (m \neq n) \end{cases} \tag{1} \\ \int_{-\pi}^\pi \sin(mx) \sin(nx) dx &= \begin{cases} \pi,\, (m = n) \\ 0,\, (m \neq n) \end{cases} \\ \end{align*}$$

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Proof of (1).

When $m = n$,

$$\begin{align*} \int_{-\pi}^\pi \cos(mx) \cos(mx) dx &= \int_{-\pi}^\pi \cos^2(mx) dx \\ &= \int_{-\pi}^\pi \frac{1 + \cos 2 m x}{2} dx \\ &= \frac{1}{2} \left[ x \right]_{-\pi}^\pi \\ &= \pi ,\\ \end{align*}$$

when $m \neq n$,

$$\begin{align*} \int_{-\pi}^\pi \cos(mx) \cos(nx) dx &= \int_{-\pi}^\pi \frac{1}{2} \left( \cos(m+n)x + \cos(m - n)x \right) dx \\ &= 0 . \\ \end{align*}$$