Trigonometry

Summary

\sin θ = \frac{o p p o s i t e}{h y p o t e n u s e}

\cos θ = \frac{a d j a c e n t}{h y p o t e n u s e}

\tan θ = \frac{o p p o s i t e}{a d j a c e n t} = \frac{\sin θ}{\cos θ}

\sin^{2} θ + \cos^{2} θ = 1

\frac{\sin α}{a} = \frac{\sin β}{b} = \frac{\sin γ}{c}

a^{2} = b^{2} + C^{2} - 2 b c \cos α

\sin (α \pm β) = \sin α \cos β \pm \cos α \sin β

\cos (α \pm β) = \cos α \cos β \pm \sin α \sin β

\tan (α \pm β) = \frac{\tan α \pm \tan β}{1 \pm \tan α \tan β}

\sin 2 α = 2 \sin α \cos α

\begin{aligned} \cos 2 α & = \cos^{2} α - \sin^{2} α \\ = 2 \cos^{2} θ - 1 \\ = 1 - 2 \sin^{2} θ \end{aligned}

\tan 2 α = \frac{2 \tan α}{1 - \tan^{2} α}

When there is an expression involving sum of sine and cosine terms:

a \sin θ \pm b \cos θ = R \sin (θ \pm α)

Where $a, b, R$ and $α$ are +ve constants,

R = \sqrt{a^{2} + b^{2}}

α = \tan^{- 1} \frac{b}{a}

Degrees	0	30	45	60	90
Radians	$0$	$\frac{π}{6}$	$\frac{π}{4}$	$\frac{π}{3}$	$\frac{π}{2}$
$\sin θ$	$0$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$	1
$\cos θ$	1	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	0
$\tan θ$	0	$\frac{1}{\sqrt{3}}$	1	$\sqrt{3}$	undefined
$\csc θ$	undefined	2	$\sqrt{2}$	$\frac{2}{\sqrt{3}}$	1
$\sec θ$	1	$\frac{2}{\sqrt{3}}$	$\sqrt{2}$	2	undefined
$\cot θ$	undefined	$\sqrt{3}$	1	$\frac{1}{\sqrt{3}}$	0

$π r a d = 180^{\circ}$
$1 r a d = \frac{180}{π}$

$x r a d = {(x \times \frac{180}{π})}^{\circ}$

$90^{\circ} = \frac{π}{2} r a d$
$180^{\circ} = π r a d$
$270^{\circ} = \frac{3 π}{2} r a d$
$360^{\circ} = 2 π r a d$

$x^{\circ} = x \times \frac{π}{180} r a d$

\csc θ = \frac{1}{\sin θ}

\sec θ = \frac{1}{\cos θ}

\cot θ = \frac{1}{\tan θ}

\tan θ = \frac{\sin θ}{\cos θ}