# Math2.org Math Tables: Trigonometric Identities

(Math)

 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)

 sin2(x) + cos2(x) = 1 tan2(x) + 1 = sec2(x) cot2(x) + 1 = csc2(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) = cos2(x) - sin2(x) = 2 cos2(x) - 1 = 1 - 2 sin2(x)

tan(2x) = 2 tan(x) / (1 - tan2(x))

sin2(x) = 1/2 - 1/2 cos(2x)

cos2(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 Angles
angle 0 30 45 60 90
sin2(a) 0/4 1/4 2/4 3/4 4/4
cos2(a) 4/4 3/4 2/4 1/4 0/4
tan2(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 oppositite B, c opposite C:

a/sin(A) = b/sin(B) = c/sin(C) (Law of Sines)

 c2 = a2 + b2 - 2ab cos(C) b2 = a2 + c2 - 2ac cos(B)a2 = b2 + c2 - 2bc cos(A)
(Law of Cosines)

(a - b)/(a + b) = tan 1/2(A-B) / tan 1/2(A+B) (Law of Tangents)