Trigonometry

### Trigonometric Table

It will be useful to memorize $\sin \theta,\cos \theta,\tan \theta$ values for $\theta = 0,\pi/3,\pi/4,\pi/2$ and $\pi \pm \theta$, $2\pi - \theta$ for the above values of $\theta$. The values of $\sin \theta$ and $\cos \theta$ for these values are given below:

$\theta$ $\sin \theta$ $\cos \theta$ $\tan \theta$
0 0 1 0
$\pi/6 = 30^{\circ}$ $\frac{1}{2}$ $\frac{\sqrt{3}}{2}$ $\frac{1}{\sqrt{3}}$
$\pi/4 = 45^{\circ}$ $\frac{1}{\sqrt{2}}$ $\frac{1}{\sqrt{2}}$ 1
$\pi/3 = 60^{\circ}$ $\frac{\sqrt{3}}{2}$ $\frac{1}{2}$ $\sqrt{3}$
$\pi/2 = 90^{\circ}$ 1 0 $\infty$

### Identities

• The following identities involving sine and cosine functions will be useful:

$\sin(\theta \pm \phi) = \sin \theta \ \cos \phi \pm \cos \theta \ \sin \phi$

$\cos(\theta \pm \phi) = \cos \theta \ \cos \phi \mp \sin \theta \ \sin \phi$

$\sin \theta \sin \phi = \frac12 \left[\cos(\theta-\phi) - \cos(\theta + \phi) \right]$

$\cos \theta \cos \phi = \frac12 \left[\cos(\theta-\phi) + \cos(\theta + \phi) \right]$

$\sin \theta \cos \phi = \frac12 \left[\sin(\theta-\phi) + \sin(\theta + \phi) \right]$

• The following special case of the above formulas are also very useful to commit to memory:

$\sin(\pi/2 \pm \phi) = \cos \phi$

$\cos(\pi/2 \pm \phi) = \mp \sin \phi$

$\sin(\pi \pm \phi) = \mp \sin \phi$

$\cos(\pi \pm \phi) = - \cos \phi$

$\cos^2 \theta = \frac12 \left(1+\cos 2 \theta\right)$

$\sin^2 \theta = \frac12 \left(1-\cos 2 \theta \right)$

$\cos(2 \theta) = 2 \cos^2 \theta - 1 = 1 - 2 \sin^2 \theta$

• Any two numbers $a$ and $b$ can be written as $r \cos \theta$ and $r \sin \theta$, where $r = \sqrt{a^2 + b^2}$ and $\theta = \tan^{-1}\left(\frac{b}{a} \right)$.
• The cosine, sine and exponential functions have infinite series (Maclaurin's series) expansions given by:

$\cos\theta = 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \frac{\theta^6}{6!} + \frac{\theta^8}{8!} - \ldots$

$\sin \theta = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \frac{\theta^7}{7!} + \frac{\theta^9}{9!} - \ldots$

$e^\theta = 1 + \frac{\theta}{1!} + \frac{\theta^2}{2!} + \frac{\theta^3}{3!} + \frac{\theta^4}{4!} + \frac{\theta^5}{5!} + \ldots$

where $\theta$ is in radians.