Trigonometric 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 y) = sin x cos y cos x sin y



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







tan(x y) = (tan x tan y) / (1 tan x 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 )



Trig Table of Common Anglesangle 0 30 45 60 90

sin^2(a) 0/4 1/4 2/4 3/4 4/4

cos^2(a) 4/4 3/4 2/4 1/4 0/4

tan^2(a) 0/4 1/3 2/2 3/1 4/0



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) (Law of Sines)

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)



(Law of Cosines)





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