Formulas & identities

Angles in radians

Key angles in radians:

$ 90^\circ = $ $ \phantom{.} {\pi \over 2} $ $ \text{ radians}$, $ 180^\circ = $ $ \phantom{.} \pi $ $ \text{ radians}$, $ 270^\circ = $ $ \phantom{.} {3\pi \over 2} $ $ \text{ radians}$, $ 360^\circ = $ $ \phantom{.} 2 \pi $ $ \text{ radians}$

Special angles in radians:

$ 30^\circ = $ $ \phantom{.} {\pi \over 6} $ $ \text{ radians}$, $ 45^\circ = $ $ \phantom{.} {\pi \over 4} $ $ \text{ radians}$, $ 60^\circ = $ $ \phantom{.} {\pi \over 3} $ $ \text{ radians}$

Trigonometric ratios

TOA, CAH, SOH:

$ \tan \theta = $ $ \phantom{.} {Opp \over Adj} $, $ \cos \theta = $ $ \phantom{.} {Adj \over Hyp} $, $ \sin \theta = $ $ \phantom{.} {Opp \over Hyp} $

Special angles:

	$30^\circ \text{ or } {\pi \over 6}$	$45^\circ \text{ or } {\pi \over 4}$	$60^\circ \text{ or } {\pi \over 3}$
$\sin \theta$	$ \phantom{.} {1 \over 2} $	$ \phantom{.} {1 \over \sqrt{2}} $	$ \phantom{.} {\sqrt{3} \over 2} $
$\cos \theta$	$ \phantom{.} {\sqrt{3} \over 2} $	$ \phantom{.} {1 \over \sqrt{2}} $	$ \phantom{.} {1 \over 2} $
$\tan \theta$	$ \phantom{.} {1 \over \sqrt{3}} $	$ \phantom{.} 1 $	$ \phantom{.} \sqrt{3} $

Maximum value, minimum value and period

For $y = a \sin bx$:

Maximum value $=$ $ \phantom{.} a $, minimum value $=$ $ \phantom{.} -a $ and period of $ {360^\circ \over b} = {2\pi \over b} $

For $y = a \cos bx$:

Maximum value $=$ $ \phantom{.} a $, minimum value $=$ $ \phantom{.} -a $ and period of $ {360^\circ \over b} = {2\pi \over b} $

For $y = a \tan bx$:

Period of $ {180^\circ \over b} = {\pi \over b} $

Identities

cot θ, sec θ, cosec θ:

(Hint: look at the third letter of each term)

$ \cot \theta = $ $ \phantom{.} {1 \over \tan \theta} $, $ \sec \theta = $ $ \phantom{.} {1 \over \cos \theta} $, $ \text{cosec } \theta = $ $ \phantom{.} {1 \over \sin \theta} $

sin θ, cos θ and tan θ:

$ \tan \theta = $ $ \phantom{.} {\sin \theta \over \cos \theta} $, $ \cot \theta = $ $ \phantom{.} {1 \over \tan \theta} = {\cos \theta \over \sin \theta} $

Pythagoreans identity (provided):

\begin{align*} \sin^2 A & + \cos^2 A = 1 \\ \sec^2 A & = 1 + \tan^2 A \\ \text{cosec}^2 A & = 1 + \cot^2 A \end{align*}

Formulas

Addition formulas (provided):

\begin{align*} \sin (A \pm B) & = \sin A \cos B \pm \cos A \sin B \\ \cos (A \pm B) & = \cos A \cos B \mp \sin A \sin B \\ \tan (A \pm B) & = { \tan A \pm \tan B \over 1 \mp \tan A \tan B} \end{align*}

Double angle formulas (provided):

\begin{align*} \sin 2A & = 2 \sin A \cos A \\ \cos 2A = \cos^2 A - \sin^2 A & = 2 \cos^2 A - 1 = 1 - 2 \sin^2 A \\ \tan 2A & = {2 \tan A \over 1 - \tan^2 A} \end{align*}

R-formulas:

$a \sin x \pm b \cos x = $ $ R \sin (x \pm \alpha) $

$a \cos x \pm b \sin x = $ $ R \cos (x \mp \alpha) $

$ \text{For both, } R = $ $\sqrt{a^2 + b^2}$ and $\alpha = $ $ \tan^{-1} \left(b \over a\right) $

Negative angles:

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

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

$ \tan (-\theta) = $ $- \tan \theta$

Deriving by addition formulas

Complementary angles (i.e. both angles add up to 90°):

$ \sin (90^\circ -\theta) = $ $ \cos \theta$

$ \cos (90^\circ -\theta) = $ $\sin \theta$