Trigonometric Identities

  

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

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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 Angles

angle	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

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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)