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Trigonometry – Quick Reference Sheet

Trigonometric Functions

Identities

$\sin(\theta)=\frac{opposite}{hypotenuse}=\frac{y}{r}$	$\csc(\theta)=\frac{hypotenuse}{opposite}=\frac{r}{y}=\frac{1}{\sin(\theta)}$
$\cos(\theta)=\frac{adjacent}{hypotenuse}=\frac{x}{r}$	$\sec(\theta)=\frac{hypotenuse}{adjacent}=\frac{r}{x}=\frac{1}{\cos(\theta)}$
$\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}=\frac{opposite}{adjacent}=\frac{y}{x}$	$\cot(\theta)=\frac{adjacent}{opposite}=\frac{x}{y}=\frac{1}{tan(\theta)}$

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

Trigonometric Function Graphs

Special Right Triangles

Pythagorean Theorem

Pythagorean Theorem
$c^2=a^2+b^2$

Converse of the Pythagorean Theorem: – If $c^2=a^2+b^2$ , then triangle ABC is a right triangle; – If $c^2 < a^2+b^2$ , then triangle ABC is an acute triangle; – If $c^2 > a^2+b^2$ , then triangle ABC is an obtuse triangle

Angle	$\boldmath{\sin(\theta)}$	$\boldmath{\cos(\theta)}$	$\boldmath{\tan(\theta)}$	$\boldmath{\csc(\theta)}$	$\boldmath{\sec(\theta)}$	$\boldmath{\cot(\theta)}$
$0^{\circ}$	0	1	0	Undefined	1	Undefined
$30^{\circ}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$	2	$\frac{2\sqrt{3}}{3}$	$\sqrt{3}$
$45^{\circ}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	1	$\sqrt{2}$	$\sqrt{2}$	1
$60^{\circ}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$	$\frac{2\sqrt{3}}{3}$	2	$\frac{\sqrt{3}}{3}$
$90^{\circ}$	1	0	Undefined	1	Undefined	0
$120^{\circ}$	$\frac{\sqrt{3}}{2}$	$\frac{-1}{2}$	$-\sqrt{3}$	$\frac{2\sqrt{3}}{3}$	-2	$\frac{-\sqrt{3}}{3}$
$135^{\circ}$	$\frac{\sqrt{2}}{2}$	$\frac{-\sqrt{2}}{2}$	$-1$	$\sqrt{2}$	$-\sqrt{2}$	-1
$150^{\circ}$	$\frac{1}{2}$	$\frac{-\sqrt{3}}{2}$	$\frac{-\sqrt{3}}{3}$	2	$\frac{-2\sqrt{3}}{3}$	$-\sqrt{3}$
$180^{\circ}$	0	-1	0	Undefined	-1	Undefined
$210^{\circ}$	$\frac{-1}{2}$	$\frac{-\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$	$-2$	$\frac{-2\sqrt{3}}{3}$	$\sqrt{3}$
$225^{\circ}$	$\frac{-\sqrt{2}}{2}$	$\frac{-\sqrt{2}}{2}$	$1$	$-\sqrt{2}$	$-\sqrt{2}$	1
$240^{\circ}$	$\frac{-\sqrt{3}}{2}$	$\frac{-1}{2}$	$\sqrt{3}$	$\frac{-2\sqrt{3}}{3}$	-2	$\frac{\sqrt{3}}{3}$
$270^{\circ}$	-1	0	Undefined	-1	Undefined	0
$300^{\circ}$	$\frac{-\sqrt{3}}{2}$	$\frac{1}{2}$	$-\sqrt{3}$	$\frac{-2\sqrt{3}}{3}$	2	$\frac{-\sqrt{3}}{3}$
$315^{\circ}$	$\frac{-\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	-1	$-\sqrt{2}$	$\sqrt{2}$	$-1$
$330^{\circ}$	$\frac{-1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{-\sqrt{3}}{3}$	$-2$	$\frac{2\sqrt{3}}{3}$	$-\sqrt{3}$
$360^{\circ}$	0	1	0	Undefined	1	Undefined

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Sum and Difference Formulas

Law of Sines

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

Law of Cosines
$\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}$
$c^2=a^2+b^2-2a \cdot b \cdot cos(C)$
Heron’s Formula
Area for any triangle with sides the lengths a,b, and c $Area=\sqrt{s(s-a)(s-b)(s-c)}$
Radian and Degree Conversion Formula
$\frac{degrees}{180}=\frac{radians}{\pi}$

Sum to Product Formulas

Radian and Degree Conversion Circle

$\sin(A)+\sin(B)=2\sin(\frac{(A+B)}{2})\cos(\frac{(A-B)}{2})$
$\sin(A)-\sin(B)=2\cos(\frac{(A+B)}{2})\sin(\frac{(A-B)}{2})$
$\cos(A)+\cos(B)=2\cos(\frac{(A+B)}{2})\sin(\frac{(A-B)}{2})$
$\cos(A)-\cos(B)=2\sin(\frac{(A+B)}{2})\sin(\frac{(A-B)}{2})$

Product to Sum Formulas
$\sin(A)\cos(B)=\frac{1}{2}[\sin(A+B)+\sin(A-B)]$ $\cos(A)\cos(B)=\frac{1}{2}[\cos(A-B)+\cos(A+B)]$ $\sin(A)\sin(B)=\frac{1}{2}[\cos(A-B)-\cos(A+B)]$ $\cos(A)\sin(B)=\frac{1}{2}[\sin(A+B)-\sin(A-B)]$

Square Formulas

Coordinate and Degree Conversion Circle

Double Angle Formulas
$\sin^2(A)=\frac{1-\cos(2A)}{2}$ $\tan^2(A)=\frac{1-\cos(2A)}{1+\cos(2A)}$ $\cos^2(A)=\frac{1+\cos(2A)}{2}$
$\sin(2A)=2\sin(A)\cos(A)$ $\tan(2A)=\frac{2\tan(A)}{1-\tan^2(A)}$ $\cos(2A)=1-2\sin^2(A)=2\cos^2(A)-1=\cos^2(A)-\sin^2(A)$
Half Angle Formulas
$\sin(\frac{A}{2})=\pm\sqrt{\frac{1-\cos(A)}{2}}$ $\cos(\frac{A}{2})=\pm\sqrt{\frac{1+\cos(A)}{2}}$ $\tan(\frac{A}{2})=\pm \sqrt{\frac{1-\cos(A)}{1+\cos(A)}}=\frac{\sin(A)}{1+\cos(A)}=\frac{1-\cos(A)}{\sin(A)}$

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