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Trig 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

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

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

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

Different Angle

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

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

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

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

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

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

Double Angle

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

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

$${2 sin(a) cos(a) = sin(2a)}$$

$${cos^2(a) - sin^2(a) = cos(2a)}$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Integrals

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Hyperbolic Trig Functions

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

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

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

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

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

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

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

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

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

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

Complex Trig Functions

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

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

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