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Math formulas: Trigonometry identities

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Right-Triangle Definitions

Right-Triangle Definition
sinα=OppositeHypotenuse \sin \alpha = \frac{\text{Opposite}}{\text{Hypotenuse}}
cosα=AdjacentHypotenuse \cos \alpha = \frac{\text{Adjacent}}{\text{Hypotenuse}}
tanα=OppositeAdjacent \tan \alpha = \frac{\text{Opposite}}{\text{Adjacent}}
cscα=1sinα=HypotenuseOpposite \csc \alpha = \frac{1}{\sin\alpha} = \frac{\text{Hypotenuse}}{\text{Opposite}}
secα=1cosα=HypotenuseAdjacent \sec \alpha = \frac{1}{\cos\alpha} = \frac{\text{Hypotenuse}}{\text{Adjacent}}
cotα=1tanα=AdjacentOpposite \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(π2x)=cos(x) \sin\left(\frac{\pi}{2} - x\right) = \cos(x)
cos(π2x)=sin(x) \cos\left(\frac{\pi}{2} - x\right) = \sin(x)
sin(π2+x)=cos(x) \sin\left(\frac{\pi}{2} + x\right) = \cos(x)
cos(π2+x)=sin(x) \cos\left(\frac{\pi}{2} + x\right) = -\sin(x)
sin(πx)=sin(x) \sin(\pi - x) = \sin(x)
cos(πx)=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

sin2x+cos2x=1 \sin^2x + \cos^2x = 1
tan2x+1=1cos2x \tan^2x + 1 = \frac{1}{\cos^2x}
cot2x+1=1sin2x \cot^2x + 1 = \frac{1}{\sin^2x}

Sum and Difference Formulas

sin(α+β)=sinαcosβ+sinβcosα \sin(\alpha + \beta) = \sin\alpha \cdot \cos \beta + \sin\beta \cdot \cos\alpha
sin(αβ)=sinαcosβsinβcosα \sin(\alpha - \beta) = \sin\alpha \cdot \cos \beta - \sin \beta \cdot \cos\alpha
cos(α+β)=cosαcosβsinαcosβ \cos(\alpha + \beta) = \cos\alpha \cdot \cos \beta - \sin\alpha \cdot \cos\beta
cos(αβ)=cosαcosβ+sinαcosβ \cos(\alpha - \beta) = \cos\alpha \cdot \cos \beta + \sin\alpha \cdot \cos\beta
tan(α+β)=tanα+tanβ1tanαtanβ \tan(\alpha + \beta) = \frac{ \tan\alpha + \tan\beta}{1 - \tan\alpha \cdot \tan\beta }
tan(αβ)=tanαtanβ1+tanαtanβ \tan(\alpha - \beta) = \frac{ \tan\alpha - \tan\beta}{1 + \tan\alpha \cdot \tan\beta }

Double Angle and Half Angle Formulas

sin(2α)=2sinαcosα \sin(2\,\alpha) = 2 \cdot \sin\alpha \cdot \cos\alpha
cos(2α)=cos2αsin2α \cos(2\,\alpha) = \cos^2\alpha - \sin^2\alpha
tan(2α)=2tanα1tan2α \tan(2\,\alpha) = \frac{2\,\tan\alpha}{1 - \tan^2\alpha}
sinα2=±1cosα2 \sin \frac{\alpha}{2} = \pm \sqrt{\frac{1-\cos\alpha}{2}}
cosα2=±1+cosα2 \cos \frac{\alpha}{2} = \pm \sqrt{\frac{1+\cos\alpha}{2}}
tanα2=1cosαsinα=sinα1cosα\tan \frac{\alpha}{2} = \frac{1 - \cos\alpha}{\sin\alpha} = \frac{\sin\alpha}{1 - \cos\alpha}
tanα2=±1+cosα1cosα\tan \frac{\alpha}{2} = \pm \sqrt{\frac{1 + \cos\alpha}{1 - \cos\alpha} }

Other Useful Trig Formulas

Law of sines

sinαα=sinββ=sinγγ \frac{\sin\alpha}{\alpha} = \frac{\sin\beta}{\beta} = \frac{\sin\gamma}{\gamma}

Law of cosines

a2=b2+c22bccosαb2=a2+c22accosβc2=a2+b22abcosγ \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=12absinγ A = \frac{1}{2} a\,b\, \sin\gamma

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