Trigonometry

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Exact Values

Radians ${(}^c)$

	$0^c$	$\frac{\pi }{6}$	$\frac{\pi }{4}$	$\frac{\pi }{3}$	$\frac{\pi }{2}$
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

Degrees (˚)

	0˚	30˚	45˚	60˚	90˚
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

Radians

One radian (written $1^c$ ) is the angle formed at the centre of the unit circle by an arc of length 1 unit $1^c=\frac{180˚}{\pi }$

$1˚=\frac{{\pi }^c}{180}$

Sin, Cosine and Tangent

Cosine

• x-coordinate of $P(\theta )$ on unit circle

$x=\cos (\theta ),\theta \in \mathbb{R}$

Sine

• y-coordinate of $P(\theta )$ on unit circle

$y=\sin (\theta ),\theta \in \mathbb{R}$

Tangent

$\tan (\theta )=\frac{\sin (\theta )}{\cos (\theta )},\cos (\theta )\ne 0$

Symmetry Properties of Trigonometric Functions

• In quadrant 1, all functions are positive

$\sin (\theta )=b$

$\cos (\theta )=a$

$\tan (\theta )=\frac{b}{a}=c$

• In quadrant 2, only sin is positive

$\sin (\theta -\pi )=b=\sin (\theta )$

$\cos (\theta -\pi )=-a=-\cos (\theta )$

$\tan (\theta -\pi )=\frac{b}{-a}=-c=\sin (\theta )$

• In quadrant 3, only tan is positive

$\sin (\theta +\pi )=-b=-\sin (\theta )$

$\cos (\theta +\pi )=-a=-\cos (\theta )$

$\tan (\theta +\pi )=\frac{-b}{-a}=c=\tan (\theta )$

• In quadrant 4, only cos is positive

$\sin (2\pi -\theta )=-b=-\sin (\theta )$

$\cos (2\pi -\theta )=a=\cos (\theta )$

$\tan (2\pi -\theta )=\frac{-b}{a}=-c=-\tan (\theta )$

Negative Angles

$\sin (-\theta )=-\sin (\theta )$

$cos(-\theta )=\cos (\theta )$

$\tan (-\theta )=\frac{-\sin (\theta )}{\cos (\theta )}=-\tan (\theta )$

Complementary Relationships

$\sin (\frac{\pi }{2}-\theta )=\cos (\theta )$

$\sin (\frac{\pi }{2}+\theta )=\cos (\theta )$

$\cos (\frac{\pi }{2}-\theta )=\sin (\theta )$

$\cos (\frac{\pi }{2}+\theta )=-\sin (\theta )$

Pythagorean Identity

• you won’t be asked to prove this (I think)

${\cos }^2(x)+{\sin }^2(x)=1$

Addition Formulas

$\cos (A+B)=\cos A\cos B-\sin A\sin B$

$\cos (A-B)=\cos A\cos B+\sin A\sin B$

$\sin (A+B)=\sin A\cos B+\cos A+\sin B$

$\sin (A-B)=\sin A\cos B-\cos A+\sin B$

$\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$

$\tan (A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}$

Sine Rule and Cosine rule

• 

for the above triangle: $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$

$c^2=a^2+b^2-2ab\cos C$

Arcs, Segments and Sectors

• These are provided on your formula sheet, but i’ll provide a brief run-down
• 

In a circle of radius $r$, for an arc subtending angle $\theta$ (radians) at the centre

Length of Arc