Trigonometric identities
We use the following standard notation: sin2 x (cos2 x, etc.) means (sin x)2
((cos x)2 , etc.).
The following identities will often be used in this course:
sin x = − sin(−x) = cos(π/2 − x) = sin(π − x) = − sin(π + x) = sin(2π + x)
cos x = cos(−x) = sin(π/2 − x) = − cos(π − x) = − cos(π + x) = cos(2π + x)
tan x = − tan(−x) = 1/ tan(π/2 − x) = − tan(π − x) = tan(π + x)
sin x
tan x =
cos x
1
cos x
=
cot x =
sin x
tan x
sec x = 1/ cos x
csc x = 1/ sin x
sin2 x + cos2 x = 1
1 + tan2 x = sec2 x
1 + cot2 x = csc2 x
sin(2x) = 2 sin x cos x
cos(2x) = cos2 x − sin2 x = 2 cos2 x − 1 = 1 − 2 sin2 x
sin(x + y) = sin x cos y + cos x sin y
sin(x − y) = sin x cos y − cos x sin y
cos(x + y) = cos x cos y − sin x sin y
cos(x − y) = cos x cos y + sin x sin y
Some other identities that can be obtained from the identities above:
cos(x − y) − cos(x + y)
2
cos(x + y) + cos(x − y)
cos x cos y =
2
sin(x + y) + sin(x − y)
sin x cos y =
2
x+y
x−y
sin x + sin y = 2 sin
cos
2
2
x−y
x+y
sin x − sin y = 2 sin
cos
2
2
x+y
x−y
cos x + cos y = 2 cos
cos
2
2
x+y
x−y
cos x − cos y = −2 sin
sin
2
2
sin x sin y =
1