Trigonometry

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:

(1)

\begin{align} \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) \\ \sin(2 \theta) & = 2 \sin \theta \cos \theta \\ \cos(2 \theta) & = 2 \cos^2 \theta - 1 = 1 - 2 \sin^2 \theta \end{align}

• Any two numbers $a$ and $b$ can be written as $a = r \cos \theta$ and $b = r \sin \theta$, 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.

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	$\infty$