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Trigonometry Cheatsheet

Definitions, Symbols, Formulas, and Notes — All in One Place.

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Trigonometric Functions

Term		Definition		Formula		Domain		Range
Sine		In a right triangle, the ratio of the length of the side opposite the angle to the length of the hypotenuse.		$\sin (θ) = \frac{Opposite}{Hypotenuse}$		$(- \infty, \infty)$		$[- 1, 1]$
Cosine		In a right triangle, the ratio of the length of the adjacent side to the length of the hypotenuse		$\cos (θ) = \frac{Adjacent}{Hypotenuse}$		$(- \infty, \infty)$		$[- 1, 1]$
Tangent		In a right triangle, the ratio of the length of the side opposite the angle to the length of the adjacent side.		$\tan (θ) = \frac{Opposite}{Adjacent}$		$\frac{π}{2} + π n, n is an integer$		$(- \infty, \infty)$
Secant		The reciprocal of the cosine of an angle.		$\sec (θ) = \frac{1}{\cos (θ)}$		$\frac{π}{2} + π n, n is an integer$		$[- 1, 1]$
Cotangent		The reciprocal of the tangent of an angle.		$\cot (θ) = \frac{1}{\tan (θ)}$		$π n, n is an integer$		$(- \infty, \infty)$
Cosecant		The reciprocal of the sine of an angle.		$\csc (θ) = \frac{1}{\sin (θ)}$		$π n, n is an integer$		$[- 1, 1]$
Radian		A unit of angular measure defined as the angle subtended by an arc of a circle that has the same length as the radius of the circle. One radian is equal to approximately 57.29 degrees.		$π$

Inverse Trigonometric Functions

Term		Definition		Domain		Range
$\arcsin (x)$ $\sin^{- 1} (x)$		The inverse of the sine function; it returns the value $y$ of the sine of $x$ .		$[- 1, 1]$		$[- \frac{π}{2}, \frac{π}{2}]$
$\arccos (x)$ $\cos^{- 1} (x)$		The inverse of the cos function; it returns the value $y$ of the cos of $x$ .		$[- 1, 1]$		$[0, π]$
$\arctan (x)$ $\tan^{- 1} (x)$		The inverse of the tangent function; it returns the value $y$ of the tan of $x$ .		$(- \infty, \infty)$		$[- \frac{π}{2}, \frac{π}{2}]$
$\sec^{- 1} (x)$		The inverse of the sec function; it returns the value $y$ of the sec of $x$ .		$(- \infty, - 1) \cup (1, \infty)$		$[0, π], y \neq \frac{π}{2}$
$\csc^{- 1} (x)$		The inverse of the csc function; it returns the value $y$ of the csc of $x$ .		$(- \infty, - 1] \cup [1, \infty)$		$[- \frac{π}{2}, \frac{π}{2}], y \neq 0$
$\cot^{- 1} (x)$		The inverse of the cot function; it returns the value $y$ of the cot of $x$ .		$(- \infty, \infty)$		$[0, π]$

Special Values for Trigonometric Functions

Function		0°		30°		45°		60°		90°
sin		$0$		$\frac{1}{2}$		$\frac{1}{\sqrt{2}}$		$\frac{\sqrt{3}}{2}$		$1$
cos		$1$		$\frac{\sqrt{3}}{2}$		$\frac{1}{\sqrt{2}}$		$\frac{1}{2}$		$0$
tan		$0$		$\frac{\sqrt{3}}{3}$		$1$		$\sqrt{3}$		Not defined
cot		Not defined		$\sqrt{3}$		$1$		$\frac{\sqrt{3}}{3}$		$0$
csc		Not defined		$2$		$\sqrt{2}$		$\frac{2 \sqrt{3}}{3}$		$1$
sec		$1$		$\frac{2 \sqrt{3}}{3}$		$\sqrt{2}$		$2$		Not defined

Trigonometric Identities

Term		Definition
Pythagorean Identities		$\sin^{2} (θ) + \cos^{2} (θ) = 1$ $1 + \tan^{2} (θ) = \sec^{2} (θ)$ $1 + \cot^{2} (θ) = \csc^{2} (θ)$
Sum and Difference Identities		$\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B$ $\cos (A \pm B) = \cos A \cos B \mp \sin A \sin B$ $\tan (A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$
Double Angle Identities		$\sin (2 θ) = 2 \sin θ \cos θ$ $\cos (2 θ) = \cos^{2} θ - \sin^{2} θ$ $\tan (2 θ) = \frac{2 \tan θ}{1 - \tan^{2} θ}$
Half Angle Identities		$\sin (\frac{θ}{2}) = \pm \sqrt{\frac{1 - \cos θ}{2}}$ $\cos (\frac{θ}{2}) = \pm \sqrt{\frac{1 + \cos θ}{2}}$ $\tan (\frac{θ}{2}) = \frac{\sin θ}{1 + \cos θ} = \frac{1 - \cos θ}{\sin θ}$
Product to Sum Identities		$\sin A \sin B = \frac{\cos (A - B) - \cos (A + B)}{2}$ $\cos A \cos B = \frac{\cos (A - B) + \cos (A + B)}{2}$ $\sin A \cos B = \frac{\sin (A + B) + \sin (A - B)}{2}$
Sum to Product Identities		$\sin A + \sin B = 2 \sin (\frac{A + B}{2}) \cos (\frac{A - B}{2})$ $\sin A - \sin B = 2 \cos (\frac{A + B}{2}) \sin (\frac{A - B}{2})$ $\cos A + \cos B = 2 \cos (\frac{A + B}{2}) \cos (\frac{A - B}{2})$ $\cos A - \cos B = - 2 \sin (\frac{A + B}{2}) \sin (\frac{A - B}{2})$
Law of Sines		$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
Law of Cosines		$c^{2} = a^{2} + b^{2} - 2 a b \cos C$ $a^{2} = b^{2} + c^{2} - 2 b c \cos A$ $b^{2} = a^{2} + c^{2} - 2 a c \cos B$

Graphs of Trigonometric Functions

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