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Trigonometric Identities
Pre-Calc
The six trigonometric functions/
functions/Reciprocals Identities
opp y
1
hyp r
1
sin x =
= =
csc x =
= =
hyp r csc x
opp y sin x
adj x
1
hyp r
1
cos x =
= =
sec x =
= =
hyp r sec x
adj x cos x
opp y sin x
adj x
1
cos x
tan x =
= =
cot x =
= =
=
adj x cos x
opp y tan x sin x
sin2x
cos2x
+
=1
tan2x – sec2x = -1
Pythagorean Identities
cos2x = 1 – sin2x
1 + tan2x = sec2x
sin2x = 1 - cos2x
1 + cot2x = csc2x
Sum or difference of two angles
sin(a + b) = sina cosb + cosa sinb
cos(a + b) = cosa cosb - sina sinb
sin(a - b) = sina cosb - cosa sinb
cos(a - b) = cosa cosb + sina sinb
tan a + tan b
tan a − tan b
tan(a + b) =
tan(a − b) =
1 − tan a tan b
1 + tan a tan b
Double Angle Formulas
cos2x = 1- 2sin2x
cos2x = 2cos2x - 1
cos2x = cos2x – sin2x
sin2x = 2sinx cosx
tan 2 x =
2 tan x
1 − tan 2 x
Reduction Formulas
sin(-x) = -sinx
sin(x) = -sin(x – π)
tan(-x) = -tanx
π

± sin x = cos x ± 
2

cos(-x) = cosx
cos(x) = -cos(x – π)
tan(x) = tan(x – π)
π

± cos x = sin  x ± 
2

Half Angle Formulas
1 − cos 2 x
1 + cos 2 x
1 − cos 2 x
sin 2 x =
cos 2 x =
tan 2 x =
2
2
1 + cos 2 x
x
1 − cos x
x
1 + cos x
sin = ±
cos = ±
2
2
2
2
x
1 − cos x
sin x
1 − cos x
tan = ±
=
=
2
1 + cos x 1 + cos x
sin x
Law of cosines
a2 = b2 + c2 – 2bc cos A
(where A is the angle of a scalene opposite side a)
sinx =
cos(90o
CoCo-Functions
– x)
cosx = sin(90 o – x)
tanx = cot(90 o – x)
secx = csc(90o – x)
Sum and Product Formulas
sina cosb = ½[sin(a + b) + sin(a - b)]
cosa cosb = ½[cos(a + b) + cos(a - b)]
cosa sinb = ½[sin(a + b) – sin(a - b)]
sina sinb = ½[cos(a – b) – cos(a + b)]
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
sin x − sin y = 2 cos
 sin 

 2   2 
x+ y x− y
cos x − cos y = −2 sin 
 sin 

 2   2 