Trig Indentities

θ is any angle, a, b, and c are sides of some triangle, and α, β, and γ are the corresponding sides of the triangle.

Law of Sines

\frac{\sin\alpha}{a}=\frac{\sin\beta}{b}=\frac{\sin\gamma}{c}

a^2 , Law of Cosines

b^2+c^2-2bc\cos\alpha

b^2 , Law of Cosines

a^2+c^2-2ac\cos\beta

c^2 , Law of Cosines

a^2+b^2-2ab\cos\gamma

\frac{a\pm b}{c} , Mollweide’s Formula

\frac{\cos\left(\frac{\alpha}{2}\pm\frac{\beta}{2}\right)}{\sin\left(\frac{\gamma}{2}\right)}

\frac{a-b}{a+b} , Law of Tangents

\frac{\tan\left(\frac{\alpha}{2}-\frac{\beta}{2}\right)}{\tan\left(\frac{\alpha}{2}+\frac{\beta}{2}\right)}

1, Pythagorean Identities

\sin^2\theta+\cos^2\theta

1, Pythagorean Identities

\sec^2\theta-\tan^2\theta

1, Pythagorean Identities

\csc^2\theta-\cot^2\theta

\sin\left(2\theta\right) , Double Angle Identity

2\sin\theta\cos\theta

\cos\left(2\theta\right) , Double Angle Identity

\cos^2\theta-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta

\tan\left(2\theta\right) , Double Angle Identity

\frac{2\tan\theta}{1-\tan^2\theta}

\csc\left(2\theta\right) , Double Angle Identity

\frac12\left(\sec\theta\csc\theta\right)

\sec\left(2\theta\right) , Double Angle Identity

\frac{\sec^2\theta}{2-\sec^2\theta}

\cot\left(2\theta\right) , Double Angle Identity

\frac{\cot^2\theta-1}{2\cot\theta}

\sin\left(3\theta\right) , Triple Angle Identity

3\sin\theta-4\sin^3\theta

\cos\left(3\theta\right) , Triple Angle Identity

4\cos^3\theta-3\cos\theta

\tan\left(3\theta\right) , Triple Angle Identity

\frac{3\tan\theta-\tan^3\theta}{1-3\tan^2\theta}

\sin\frac{\theta}{2} , Half Angle Identity

\pm\sqrt{\frac{1-\cos\theta}{2}}

\cos\frac{\theta}{2} , Half Angle Identity

\pm\sqrt{\frac{1+\cos\theta}{2}}

\tan\frac{\theta}{2} , Half Angle Identity

\pm\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}

\sin^2\theta , Power Reducing Identities

\frac{1-\cos\left(2\theta\right)}{2}

\cos^2\theta , Power Reducing Identities

\frac{1+\cos\left(2\theta\right)}{2}

\tan^2\theta , Power Reducing Identities

\frac{1-\cos\left(2\theta\right)}{1+\cos\left(2\theta\right)}

\sin\left(\alpha\pm\beta\right) , Sum and Difference Identities

\sin\alpha\cos\beta\pm\cos\alpha\sin\beta

\cos\left(\alpha\pm\beta\right) , Sum and Difference Identities

\cos\alpha\cos\beta\mp\sin\alpha\sin\beta

\tan\left(\alpha\pm\beta\right) , Sum and Difference Identities

\frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\tan\beta}

\sin\alpha\sin\beta , Product to Sum Identities

\frac{1}2[\cos(\alpha-\beta)-\cos(\alpha+\beta)]

\cos\alpha\cos\beta , Product to Sum Identities

\frac{1}2[\cos(\alpha-\beta)+\cos\alpha+\beta)]

\sin\alpha\cos\beta , Product to Sum Identities

\frac{1}2[\sin(\alpha+\beta)+\sin(\alpha-\beta)]

\cos\alpha\sin\beta , Product to Sum Identities

\frac{1}2[\sin(\alpha+\beta)-\sin(\alpha-\beta)]

\sin\alpha+\sin\beta , Sum to Product Identities

2\sin\bigg(\frac{\alpha+\beta}{2}\bigg)\cos\left(\frac{\alpha-\beta}{2}\right)

\sin\alpha-\sin\beta , Sum to Product Identities

2\cos\bigg(\frac{\alpha+\beta}{2}\bigg)\sin\bigg(\frac{\alpha-\beta}{2}\bigg)

\cos\alpha+\cos\beta , Sum to Product Identities

2\cos\bigg(\frac{\alpha+\beta}{2}\bigg)\cos\bigg(\frac{\alpha-\beta}{2}\bigg)

\cos\alpha-\cos\beta , Sum to Product Identities

-2\sin\bigg(\frac{\alpha+\beta}{2}\bigg)\sin\bigg(\frac{\alpha-\beta}{2}\bigg)

\sinh x , Definition

\frac{e^{x}-e^{-x}}{2}

\cosh x , Definition

\frac{e^{x}+e^{-x}}{2}

\sin\theta , Complementary Functions

\cos\left(\frac{\pi}{2}-\theta\right)

\tan\theta , Complementary Functions

\cot\left(\frac{\pi}{2}-\theta\right)

\sec\theta , Complementary Functions

\csc\left(\frac{\pi}{2}-\theta\right)

-\sin\theta , Even and Odd Functions

\sin\left(-\theta\right)

\cos\theta , Even and Odd Functions

\cos\left(-\theta\right)

-\tan\theta , Even and Odd Functions

\tan\left(-\theta\right)

\sin\theta , Half Angle Identities

\frac{2\tan\left(\frac{\theta}{2}\right)}{1+\tan^2\left(\frac{\theta}{2}\right)}

\cos\theta , Half Angle Identities

\frac{1-\tan^2\left(\frac{\theta}{2}\right)}{1+\tan^2\left(\frac{\theta}{2}\right)}

\tan\theta , Half Angle Identities

\frac{2\tan\left(\frac{\theta}{2}\right)}{1-\tan^2\left(\frac{\theta}{2}\right)}