Trigonometry

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Identities
sin(theta) = a / c csc(theta) = 1 / sin(theta) = c / a

cos(theta) = b / c sec(theta) = 1 / cos(theta) = c / b

tan(theta) = sin(theta) / cos(theta) = a / b cot(theta) = 1/ tan(theta) = b / a

sin(-x) = -sin(x) csc(-x) = -csc(x) cos(-x) = cos(x) sec(-x) = sec(x) tan(-x) = -tan(x) cot(-x) = -cot(x)

sin^2(x) + cos^2(x) = 1 tan^2(x) + 1 = sec^2(x) cot^2(x) + 1 = csc^2(x)

sin(x \pm y) = sin x cos y \pm cos x sin y cos(x \pm y) = cos x cosy \pm sin x sin y

tan(x \pm y) = (tan x \pm tan y) / (1 tan x \mp tan y)

sin(2x) = 2 sin x cos x

cos(2x) = cos^2(x) - sin^2(x) = 2 cos^2(x) - 1 = 1 - 2 sin^2(x)

tan(2x) = 2 tan(x) / (1 - tan^2(x))

sin^2(x) = 1/2 - 1/2 cos(2x)

cos^2(x) = 1/2 + 1/2 cos(2x)

sin x - sin y = 2 sin( (x - y)/2 ) cos( (x + y)/2 )

cos x - cos y = -2 sin( (x - y)/2 ) sin( (x + y)/2 )

Given Triangle abc, with angles A,B,C; a is opposite to A, b opposite B, c opposite C:
a/sin(A) = b/sin(B) = c/sin(C)

c^2 = a^2 + b^2 - 2ab cos(C)

b^2 = a^2 + c^2 - 2ac cos(B)

a^2 = b^2 + c^2 - 2bc cos(A)

(a - b)/(a + b) = tan [(A-B)/2] / tan [(A+B)/2] (Law of Tangents)

Hyperbolic Definitions
sinh(x) = ( e^x - e^-x )/2

csch(x) = 1/sinh(x) = 2/( e^x - e^-x )

cosh(x) = ( e^x + e -x )/2

sech(x) = 1/cosh(x) = 2/( e^x + e^-x )

tanh(x) = sinh(x)/cosh(x) = ( e^x - e^-x )/( e^x + e^-x )

coth(x) = 1/tanh(x) = ( e^x + e^-x)/( e^x - e^-x )

cosh^2(x) - sinh^2(x) = 1

tanh^2(x) + sech^2(x) = 1

coth^2(x) - csch^2(x) = 1

Inverse Hyperbolic Definitions
arcsinh(z) = ln( z + sqrt(z^2 + 1) )

arccosh(z) = ln( z \pm sqrt(z^2 - 1) )

arctanh(z) = 1/2 ln( (1+z)/(1-z) )

arccsch(z) = ln( (1+sqrt(1+z^2) )/z )

arcsech(z) = ln( (1 \pm sqrt(1-z^2) )/z )

arccoth(z) = 1/2 ln( (z+1)/(z-1) )

Relations to Trigonometric Functions
sinh(z) = -i sin(iz)

csch(z) = i csc(iz)

cosh(z) = cos(iz)

sech(z) = sec(iz)

tanh(z) = -i tan(iz)

coth(z) = i cot(iz)