Trigonometry Identities and Formulas

Pythagorean

$${\sin^2(x) + \cos^2(x) = 1}$$

$${1 + \cot^2(x) = \csc^2(x)}$$

$${\tan^2(x) + 1 = \sec^2(x)}$$

Angle Sum

$${\sin(a+b) = \sin(a) \cos(b) + \cos(a) \sin(b)}$$

$${\cos(a+b) = \cos(a) \cos(b) - \sin(a) \sin(b)}$$

$${\tan(a+b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}}$$

$${\tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}\left(\frac{a+b}{1 - ab}\right)}$$

Angle Difference

$${\sin(a-b) = \sin(a) \cos(b) - \cos(a) \sin(b)}$$

$${\cos(a-b) = \cos(a) \cos(b) + \sin(a) \sin(b)}$$

$${\tan(a-b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)}}$$

Different Angle

$${\sin(a) \cos(b) = \frac{1}{2}(\sin(a - b) + \sin(a + b))}$$

$${\cos(a) \sin(b) = \frac{1}{2}(\sin(a + b) - \sin(a - b))}$$

$${\sin(a) \sin(b) = \frac{1}{2}(\cos(a - b) - \cos(a + b))}$$

$${\cos(a) \cos(b) = \frac{1}{2}(\cos(a - b) + \cos(a + b))}$$

Dirichlet Kernel

$${\frac{\sin(nx + \frac{x}{2})}{\sin\left(\frac{x}{2}\right)} = 1 + 2 \sum_{k=1}^{n} \cos (kx)}$$

Complementary Angles

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

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

$${\tan(x) = \cot (\frac{\pi}{2} - x)}$$

$${\cot(x) = \tan (\frac{\pi}{2} - x)}$$

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

$${\sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}}$$

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

$${-\tan(x) = \cot (\frac{\pi}{2} + x)}$$

$${-\cot(x) = \tan (\frac{\pi}{2} + x)}$$

$${\tan^{-1}(x) + \tan^{-1}\left(\frac{1}{x}\right) = \frac{\pi}{2}, x > 0}$$

$${\tan^{-1}(x) + \cot^{-1}(x) = \frac{\pi}{2}}$$

$${\tan^{-1}(-x) = -\tan^{-1}(x)}$$

$${\tan^{-1}(x) = \cot^{-1}\left(\frac{1}{x}\right), x > 0}$$

$${\cot^{-1}(x) = \tan^{-1}\left(\frac{1}{x}\right), x > 0}$$

Supplementary Angles

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

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

$${\tan (\pi - x) = - \tan(x)}$$

$${\sin (x \pm \pi) = - \sin(x)}$$

$${\cos (x \pm \pi) = - \cos(x)}$$

$${\tan (x \pm \pi) = \tan(x)}$$

Power Reduction

$${\sin^2(x) = \frac{1}{2}(1 - \cos(2x)))}$$

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

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

Double Angle

$${\sin(2x) = 2 \sin(x) \cos(x)}$$

$${\sin(2x) = (\sin(x) + \cos(x))^2 - 1}$$

$${\sin(2x) = 1 - (\sin(x) - \cos(x))^2}$$

$${\cos(2x) = \cos^2(x) - \sin^2(x)}$$

$${\cos(2x) = 1 - 2 \ \sin^2(x)}$$

$${\cos(2x) = 2 \ \cos^2(x) - 1}$$

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

Triple Angle

$${\sin(3x) = 3 \ \sin(x) - 4 \ \sin^3(x)}$$

$${\cos(3x) = 4 \ \cos^3(x) - 3 \ \cos(x)}$$

$${\tan(3x) = \frac{3 \ \tan(x) - \tan^3(x)}{1 - 3\ \tan^2(x)}}$$

Half Angle

$${\sin\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1}{2}(1 - \cos(x))}}$$

$${\cos\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1}{2}(1 + \cos(x))}}$$

$${\tan\left(\frac{x}{2}\right) = \frac{1 - \cos(x)}{\sin(x)}}$$

$${\tan\left(\frac{x}{2}\right) = \frac{\sin(x)}{1 + \cos(x)}}$$

$${\sinh\left(\frac{x}{2}\right) = + \sqrt{\frac{1}{2}(\cosh(x)-1)}, x \ge 0}$$

$${\sinh\left(\frac{x}{2}\right) = - \sqrt{\frac{1}{2}(\cosh(x)-1)}, x \lt 0}$$

$${\cosh\left(\frac{x}{2}\right) = \sqrt{\frac{1}{2}(\cosh(x) + 1)}}$$

$${\tanh\left(\frac{x}{2}\right) = + \sqrt{\frac{\cosh(x)-1}{\cosh(x)+1}}, x \ge 0}$$

$${\tanh\left(\frac{x}{2}\right) = - \sqrt{\frac{\cosh(x)-1}{\cosh(x)+1}}, x \lt 0}$$

$${\tanh\left(\frac{x}{2}\right) = \frac{\cosh(x) - 1}{\sinh(x)}, x \ne 0}$$

$${\tanh\left(\frac{x}{2}\right) = \frac{\sinh(x)}{1 + \cosh(x)}}$$

Law of Sines

$${\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}}$$

'a' is the side opposite angle A, 'b' is opposite angle B, 'c' is opposite angle C

Law of Cosines

$${c^2 = a^2 + b^2 - 2ab \ \cos(C)}$$

where c is the side opposite angle C

Derivatives

$${\frac{d}{dx} (\sin (x)) = \cos(x)}$$

$${\frac{d}{dx} (\cos (x)) = -\sin(x)}$$

$${\frac{d}{dx} (\tan (x)) = \sec^2(x)}$$

$${\frac{d}{dx} (\sec (x)) = \sec(x)\tan(x)}$$

$${\frac{d}{dx} (\cot (x)) = -\csc^2(x)}$$

$${\frac{d}{dx} (\csc (x)) = -\csc(x)\cot(x)}$$

$${\frac{d}{dx} (\sinh (x)) = \cosh(x)}$$

$${\frac{d}{dx} (\cosh (x)) = \sinh(x)}$$

$${\frac{d}{dx} (\tanh (x)) = \DeclareMathOperator{\sech}{sech}\sech^2(x)}$$

$${\frac{d}{dx} (\coth (x)) = -\DeclareMathOperator{\csch}{csch}\csch^2(x)}$$

$${\frac{d}{dx} (\DeclareMathOperator{\sech}{sech}\sech (x)) = -\DeclareMathOperator{\sech}{sech}\sech(x) \tanh(x)}$$

$${\frac{d}{dx} (\DeclareMathOperator{\csch}{csch}\csch (x)) = -\DeclareMathOperator{\csch}{csch}\csch(x) \coth(x)}$$

$${\frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{\sqrt{1 - x^2}}}$$

$${\frac{d}{dx}(\tan^{-1}(x)) = \frac{1}{1 + x^2}}$$

$${\frac{d}{dx}(\sec^{-1}(x)) = \frac{1}{x \sqrt{x^2 - 1}}}$$

$${\frac{d}{dx}(\cos^{-1}(x)) = \frac{-1}{\sqrt{1 - x^2}}}$$

$${\frac{d}{dx}(\cot^{-1}(x)) = \frac{-1}{1 + x^2}}$$

$${\frac{d}{dx}(\csc^{-1}(x)) = \frac{-1}{x \sqrt{x^2 - 1}}}$$

$${\frac{d}{dx}(\sinh^{-1}(x)) = \frac{1}{\sqrt{1 + x^2}}}$$

$${\frac{d}{dx}(\cosh^{-1}(x)) = \frac{1}{\sqrt{x^2 - 1}}}$$

$${\frac{d}{dx}(\tanh^{-1}(x)) = \frac{1}{1 - x^2}}$$

Integrals

$${\int{\sin (x) dx} = -\cos(x) + C}$$

$${\int{\cos (x) dx} = \sin(x) + C}$$

$${\int{\sec (x) dx} = \ln |\sec(x) + \tan(x)| + C}$$

$${\int{\tan (x) dx} = \ln |\sec(x)| + C}$$

$${\int{\cot (x) dx} = \ln|\sin(x)| + C}$$

$${\int{\csc (x) dx} = \ln|\csc(x) - \cot(x)| + C}$$

$${\int{\csc (x) dx} = - \ln|\csc(x) + \cot(x)| + C}$$

$${\int{\sec^2 (x) dx} = \tan(x) + C}$$

$${\int{\csc^2 (x) dx} = -\cot(x) + C}$$

$${\int{\sec(x) \tan(x) dx} = \sec(x) + C}$$

$${\int{\sec^3 (x) dx} = \frac{1}{2} \sec(x) \tan(x) + \frac{1}{2} \ln |\sec(x) + \tan(x)| + C}$$

$${\int{\csc^3 (x) dx} = -\frac{1}{2} \csc(x) \cot(x) + \frac{1}{2} \ln |\csc(x) - \cot(x)| + C}$$

$${\int{\tan^2 (x) dx} = \tan(x) - x + C}$$

$${\int{\cot^2 (x) dx} = -\cot(x) - x + C}$$

$${\int{\sinh(x) dx} = \cosh(x) + C}$$

$${\int{\cosh(x) dx} = \sinh(x) + C}$$

$${\int{\tanh(x) dx} = \ln |\cosh(x)| + C}$$

$${\int{\coth(x) dx} = \ln|\sinh(x)| + C}$$

$${\int{\DeclareMathOperator{\sech}{sech}\sech(x) dx} = \tan^{-1}(\sinh(x)) + C}$$

$${\int{\DeclareMathOperator{\sech}{sech}\sech(x) dx} = 2 \ \tan^{-1}(e^x) + C}$$

$${\int{\DeclareMathOperator{\csch}{csch}\csch(x) dx} = \ln|\coth(x) - \DeclareMathOperator{\csch}{csch}\csch(x)| + C}$$

$${\int{\tanh^2(x) dx} = x - \tanh(x) + C}$$

$${\int{\coth^2(x) dx} = x - \coth(x) + C}$$

$${\int{\DeclareMathOperator{\sech}{sech}\sech^2(x) dx} = \tanh(x) + C}$$

$${\int{\DeclareMathOperator{\csch}{csch}\csch^2(x) dx} = - \coth(x) + C}$$

$${\int{\DeclareMathOperator{\sech}{sech}\sech(x) \tanh(x) dx} = -\DeclareMathOperator{\sech}{sech}\sech(x) + C}$$

$${\int{\DeclareMathOperator{\csch}{csch}\csch(x) \coth(x) dx} = -\DeclareMathOperator{\csch}{csch}\csch(x) + C}$$

$${\int{\frac{1}{a^2 + x^2} dx} = \frac{1}{a} \tan^{-1}(\frac{x}{a}) + C}$$

$${\int{\frac{1}{\sqrt{a^2 - x^2}} dx} = \sin^{-1}(\frac{x}{a}) + C}$$

$${\int{\frac{1}{x \sqrt{x^2 - a^2}} dx} = \frac{1}{a} \sec^{-1}|\frac{x}{a}| + C}$$

$${\int{\frac{1}{x^2 - a^2} dx} = -\frac{1}{a} \tanh^{-1}\left(\frac{x}{a}\right) + C}$$

$${\int{\frac{1}{x^2 - a^2} dx} = \frac{1}{2a} \ln \left\lvert \frac{x-a}{x+a} \right \rvert + C}$$

$${\int{\frac{1}{a^2 - x^2} dx} = \frac{1}{2a} \ln \left\lvert \frac{a+x}{a-x} \right \rvert + C}$$

$${\int{\frac{1}{\sqrt{1 + x^2}} dx} = \sinh^{-1}(x) + C}$$

$${\int{\frac{1}{\sqrt{x^2 - 1}} dx} = \cosh^{-1}(x) + C}$$

$${\int{\frac{1}{1 - x^2} dx} = \tanh^{-1}(x) + C}$$

Hyperbolic Trig Functions

$${\sinh(x) = \frac{e^{x} - e^{-x}}{2}}$$

$${\cosh(x) = \frac{e^{x} + e^{-x}}{2}}$$

$${\tanh(x) = \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}}$$

$${\cosh^2(x) - \sinh^2(x) = 1}$$

$${1 - \tanh^2(x) = \DeclareMathOperator{\sech}{sech}\sech^2(x)}$$

$${\coth^2(x) - 1 = \DeclareMathOperator{\csch}{csch}\csch^2(x)}$$

$${\sinh^2(x) + \cosh^2(x) = \cosh(2x)}$$

$${\sinh^2(x) = -\frac{1}{2} + \frac{1}{2}\cosh(2x)}$$

$${\cosh^2(x) = \frac{1}{2} + \frac{1}{2}\cosh(2x)}$$

$${2 \sinh(x) \cosh(x) = \sinh(2x)}$$

$${\cosh(x) + \sinh(x) = e^x}$$

$${\cosh(x) - \sinh(x) = e^{-x}}$$

$${\sinh^{-1}(x) = \ln (x + \sqrt{x^2 + 1}) }$$

$${\cosh^{-1}(x) = \ln (x + \sqrt{x^2 - 1})}$$

$${\tanh^{-1}(x) = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right)}$$

$${\tanh^{-1}\left(\frac{a}{b}x\right) = \frac{1}{2} \ln \left(\frac{b+ax}{b-ax}\right)}$$

$${\coth^{-1}(x) = \frac{1}{2} \ln \left(\frac{x+1}{x-1}\right)}$$

$${\DeclareMathOperator{\sech}{sech}\sech^{-1}(x) = \ln \left(\frac{1 + \sqrt{1 - x^2}}{x}\right), 0 \lt x \le 1}$$

$${\DeclareMathOperator{\csch}{csch}\csch^{-1}(x) = \ln \left(\frac{1 + \sqrt{1 + x^2}}{x}\right)}$$

Complex Trig Functions

$${\sin(x) = \frac{e^{ix} - e^{-ix}}{2i}}$$

$${\cos(x) = \frac{e^{ix} + e^{-ix}}{2}}$$

$${\tan(x) = \frac{e^{ix} - e^{-ix}}{i(e^{ix} + e^{-ix})}}$$

$${\tan^{-1}(x) = -i \ \tanh^{-1}(ix)}$$

$${\tan^{-1}(x) = \frac{1}{2} i \ \ln\left(\frac{1-ix}{1+ix}\right)}$$

$${\cosh(x) = \cos(it)}$$

$${\sinh(x) = -i \sin(it)}$$

Weierstrass Substitution (t-formula)

$${t = \tan(\frac{x}{2})}$$

$${\sin(x) = \frac{2t}{1+t^2}}$$

$${\cos(x) = \frac{1-t^2}{1+t^2}}$$

$${\tan(x) = \frac{2t}{1-t^2}}$$

$${dx = \frac{2}{1+t^2} dt}$$

Weierstrass Hyperbolic Substitution

$${t = \tanh(\frac{x}{2})}$$

$${\sinh(x) = \frac{2t}{1-t^2}}$$

$${\cosh(x) = \frac{1+t^2}{1-t^2}}$$

$${\tanh(x) = \frac{2t}{1+t^2}}$$

$${dx = \frac{2}{1-t^2} dt}$$