All Maths Formulas

Algebra

Functions TrigExpLogHyp

Analytic Geometry

Limits, Derivatives

Indefinite Integrals

Definite Integrals

Series Arith Geo Taylor

Trigonometry Identities

Right-Triangle Definitions

sin α = Opposite Hypotenuse

$\sin \alpha = \frac{\text{Opposite}}{\text{Hypotenuse}}$

cos α = Adjacent Hypotenuse

$\cos \alpha = \frac{\text{Adjacent}}{\text{Hypotenuse}}$

tan α = Opposite Adjacent

$\tan \alpha = \frac{\text{Opposite}}{\text{Adjacent}}$

csc α = 1 sin α = Hypotenuse Opposite

$\csc \alpha = \frac{1}{\sin\alpha} = \frac{\text{Hypotenuse}}{\text{Opposite}}$

sec α = 1 cos α = Hypotenuse Adjacent

$\sec \alpha = \frac{1}{\cos\alpha} = \frac{\text{Hypotenuse}}{\text{Adjacent}}$

cot α = 1 tan α = Adjacent Opposite

$\cot \alpha = \frac{1}{\tan\alpha} = \frac{\text{Adjacent}}{\text{Opposite}}$

Reduction Formulas

sin (- x) = - sin (x)

$\sin(-x) = -\sin(x)$

cos (- x) = cos (x)

$\cos(-x) = \cos(x)$

sin (π 2 - x) = cos (x)

$\sin\left(\frac{\pi}{2} - x\right) = \cos(x)$

cos (π 2 - x) = sin (x)

$\cos\left(\frac{\pi}{2} - x\right) = \sin(x)$

sin (π 2 + x) = cos (x)

$\sin\left(\frac{\pi}{2} + x\right) = \cos(x)$

cos (π 2 + x) = - sin (x)

$\cos\left(\frac{\pi}{2} + x\right) = -\sin(x)$

sin (π - x) = sin (x)

$\sin(\pi - x) = \sin(x)$

cos (π - x) = - cos (x)

$\cos(\pi - x) = -\cos(x)$

sin (π + x) = - sin (x)

$\sin(\pi + x) = -\sin(x)$

cos (π + x) = - cos (x)

$\cos(\pi + x) = -\cos(x)$

Basic Identities

sin 2 x + cos 2 x = 1

$\sin^2x + \cos^2x = 1$

tan 2 x + 1 = 1 cos 2 x

$\tan^2x + 1 = \frac{1}{\cos^2x}$

cot 2 x + 1 = 1 sin 2 x

$\cot^2x + 1 = \frac{1}{\sin^2x}$

Sum and Difference Formulas

sin (α + β) = sin α \cdot cos β + sin β \cdot cos α

$\sin(\alpha + \beta) = \sin\alpha \cdot \cos \beta + \sin\beta \cdot \cos\alpha$

sin (α - β) = sin α \cdot cos β - sin β \cdot cos α

$\sin(\alpha - \beta) = \sin\alpha \cdot \cos \beta - \sin \beta \cdot \cos\alpha$

cos (α + β) = cos α \cdot cos β - sin α \cdot cos β

$\cos(\alpha + \beta) = \cos\alpha \cdot \cos \beta - \sin\alpha \cdot \cos\beta$

cos (α - β) = cos α \cdot cos β + sin α \cdot cos β

$\cos(\alpha - \beta) = \cos\alpha \cdot \cos \beta + \sin\alpha \cdot \cos\beta$

tan (α + β) = tan α + tan β 1 - tan α \cdot tan β

$\tan(\alpha + \beta) = \frac{ \tan\alpha + \tan\beta}{1 - \tan\alpha \cdot \tan\beta }$

tan (α - β) = tan α - tan β 1 + tan α \cdot tan β

$\tan(\alpha - \beta) = \frac{ \tan\alpha - \tan\beta}{1 + \tan\alpha \cdot \tan\beta }$

Double Angle and Half Angle Formulas

sin (2 α) = 2 \cdot sin α \cdot cos α

$\sin(2\,\alpha) = 2 \cdot \sin\alpha \cdot \cos\alpha$

cos (2 α) = cos 2 α - sin 2 α

$\cos(2\,\alpha) = \cos^2\alpha - \sin^2\alpha$

tan (2 α) = 2 tan α 1 - tan 2 α

$\tan(2\,\alpha) = \frac{2\,\tan\alpha}{1 - \tan^2\alpha}$

sin α 2 = \pm 1 - cos α 2 - - - - - - - - \sqrt

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

cos α 2 = \pm 1 + cos α 2 - - - - - - - - \sqrt

$\cos \frac{\alpha}{2} = \pm \sqrt{\frac{1+\cos\alpha}{2}}$

tan α 2 = 1 - cos α sin α = sin α 1 - cos α

$\tan \frac{\alpha}{2} = \frac{1 - \cos\alpha}{\sin\alpha} = \frac{\sin\alpha}{1 - \cos\alpha}$

tan α 2 = \pm 1 + cos α 1 - cos α - - - - - - - - \sqrt

$\tan \frac{\alpha}{2} = \pm \sqrt{\frac{1 + \cos\alpha}{1 - \cos\alpha} }$

Other Useful Trig Formulas

Law of sines

sin α α = sin β β = sin γ γ

$\frac{\sin\alpha}{\alpha} = \frac{\sin\beta}{\beta} = \frac{\sin\gamma}{\gamma}$

Law of cosines

a 2 = b 2 + c 2 - 2 \cdot b \cdot c \cdot cos α b 2 = a 2 + c 2 - 2 \cdot a \cdot c \cdot cos β c 2 = a 2 + b 2 - 2 \cdot a \cdot b \cdot cos γ

$\begin{aligned} a^2 = b^2 + c^2 - 2\cdot b\cdot c\cdot \cos\alpha \\ b^2 = a^2 + c^2 - 2\cdot a\cdot c\cdot \cos\beta \\ c^2 = a^2 + b^2 - 2\cdot a\cdot b\cdot \cos\gamma \end{aligned}$

Area of triangle

A = 1 2 a b sin γ

$A = \frac{1}{2} a\,b\, \sin\gamma$

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