# Precalculus

Rules of Thumb

All trig functions are graphable and all can be derived from unit circle
Period of a function $\neq b$ value of a function. Period $=\frac{2\pi}{b}$
$\cos x^2\neq\cos^2x$
All angles in a triangle add up to $180$°
You cannot apply trig identities to triangles that are not RIGHT (have a $90$° angle). You need to use laws of cosines and sines to solve these
$\cos, \sin, \tan,$ etc.are not constants or variables and cannot be left by themselves without a number or variable following them. They are operators.
Capital letters such as A, B, and C usually refer to angles while lowercase letters such as a, b, and c usually refer to sides

What to Review

Trig function graphs
Memorize Unit Circle and Trig Identities

Circular Function Form

Circular functions are of the form $a\{operator\}b(x-c)+d$
Where $a$ is the amplitude, $b$ relates to the period (Note that $b$ is not the period. The period is $2b$ for most trig functions), $c$ is the horizontal shift, and $d$ is the vertical shift.
The operator can be any trigonometric operator such as $\sin$ or $\cos$

The Unit Circle

A circle with radius of 1, probably the most useful thing in precalc.
The unit circle shows ordered pairs. These ordered pairs are of the form ($\cos, \sin$).

List of Trigonometric Identities

Reciprocal Identities

$\tan\theta=\frac{\sin\theta}{\cos\theta}$
$\cot\theta=\frac{\cos\theta}{\sin\theta}$
$\sec\theta=\frac{1}{\cos\theta}$
$\csc\theta=\frac{1}{\sin\theta}$
$\tan\theta=\frac{1}{\cot\theta}$
$\cot\theta=\frac{1}{\tan\theta}$

Pythagorean Identities

$\sin^2\theta+\cos^2\theta=1$
$1+\cot^2\theta=\csc^2\theta$
$1+\tan^2\theta=\sec^2\theta$

Sum and Difference Formulas

$\cos(a\pm b)=\cos(a)\cos(b)\mp\sin(a)\sin(b)$
$\sin(a\pm b)=\sin(a)\cos(b)\pm\cos(a)\sin(b)$

Note: for the COSINE formula, the sign in the formula is the opposite of the operation which you’re doing. (e.g. $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$ not $\cos(a)\cos(b)+\sin(a)\sin(b)$)

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

Double Angle Formulas

$\sin2\theta=2\sin\theta\cos\theta$
$\cos2\theta=\cos^2\theta-\sin^2\theta$
$\cos2\theta=1-2\sin^2\theta$
$\cos2\theta=2\cos^2\theta-1$

Half Angle Formulas

$\cos\frac{\theta}{2}=\pm\sqrt{\frac{1+\cos\theta}{2}}$
$\sin\frac{\theta}{2}=\pm\sqrt{\frac{1-\cos\theta}{2}}$

Tangent Formulas

$\tan(a+b)=\frac{\tan(a)+\tan(b)}{1-\tan(a)\tan(b)}$
$\tan(a-b)=\frac{\tan(a)+\tan(b)}{1+\tan(a)\tan(b)}$
$\tan2\theta=\frac{2\tan\theta}{1-tan^2\theta}$
$\tan\frac{\theta}{2}=\frac{\sin\theta}{1+\cos\theta}$
$\tan\frac{\theta}{2}=\frac{1-\cos\theta}{\sin\theta}$
$\tan\frac{\theta}{2}=\pm\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$

Formulas for Oblique Triangles

Law of Sines

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$

Law of Cosines

$a^2=b^2+c^2-2bc(\cos A)$
$b^2=a^2+c^2-2ac(\cos B)$
$c^2=a^2+b^2-2ab(\cos C)$
$\cos A=\frac{b^2+c^2-a^2}{2bc}$
$\cos B=\frac{a^2+c^2-b^2}{2ac}$
$\cos C=\frac{a^2+b^2-c^2}{2ab}$

Area Formula for Oblique (non-right) Triangles

$\frac{1}{2}bc(\sin A)=\frac{1}{2}ac(\sin B)=\frac{1}{2} ab(\sin C)$