Basic trigonometric identities Common angles
Basic trigonometric identities Common angles
Degrees 0 30 45 60 90
Radians 0
6
4
3
2
1
2
3
sin
0
1
2
2
2
3
2
1
cos
1
0
2
2
2
3
tan
0
1
3
3
Reciprocal functions Even/odd
1 cot x =
tan x 1
csc x = sin x 1
sec x = cos x
Pythagorean identities
sin(-x) = - sin x cos(-x) = cos x tan(-x) = - tan x
sin2 x + cos2 x = 1 1 + tan2 x = sec2 x 1 + cot2 x = csc2 x
1
Cofunction identities
sin( - x) = cos x
2 cos( - x) = sin x 2 tan( - x) = cot x 2 cot( - x) = tan x 2 sec( - x) = csc x 2 csc( - x) = sec x 2
Sum and difference of angles
Double angles
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
tan x + tan y tan(x + y) =
1 - tan x tan y tan x - tan y tan(x - 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
2 tan x tan(2x) = 1 - tan2 x
2
Half angles
Power reducing formulas
x
1 - cos x
sin = ?
2
2
x
1 + cos x
cos = ?
2
2
x 1 - cos x tan =
2 sin x
sin x =
1 + cos x
Product to sum
sin2 x = 1 - cos 2x 2
cos2 x = 1 + cos 2x 2
tan2 x = 1 - cos 2x 1 + cos 2x
1 sin x sin y = cos(x - y) - cos(x + y)
2 1 cos x cos y = cos(x - y) + cos(x + y) 2 1 sin x cos y = sin(x + y) + sin(x - y) 2 tan x + tan y tan x tan y = cot x + cot y tan x + cot y tan x cot y = cot x + tan y
3
Sum to product
x+y
x-y
sin x + sin y = 2 sin
cos
2
2
x+y
x-y
sin x - sin y = 2 cos
sin
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 + y) tan x + tan y =
cos x cos y
sin(x - y) tan x - tan y =
cos x cos y
Source
Basic triginometric identities by Evgenii Neumerzhitckii is licensed under a Creative Commons Attribution 4.0 International License.
4
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