USEFUL TRIGONOMETRIC IDENTITIES - University of Adelaide
[Pages:2]USEFUL TRIGONOMETRIC IDENTITIES
1 sec x =
cos x
Definitions
sin x tan x =
cos x 1
cosec x = sin x
1 cot x =
tan x
Fundamental trig identity
(cos x)2 + (sin x)2 = 1 1 + (tan x)2 = (sec x)2 (cot x)2 + 1 = (cosec x)2
Odd and even properties cos(-x) = cos(x) sin(-x) = - sin(x) tan(-x) = - tan(x)
Double angle formulas
sin(2x) = 2 sin x cos x
cos(2x) = (cos x)2 - (sin x)2 cos(2x) = 2(cos x)2 - 1 cos(2x) = 1 - 2(sin x)2
Half angle formulas
sin(
1 2
x)
2
=
1 2
(1
- cos x)
cos(
1 2
x)
2
=
1 2
(1
+ cos x)
Sums and differences of angles cos(A + B) = cos A cos B - sin A sin B cos(A - B) = cos A cos B + sin A sin B sin(A + B) = sin A cos B + cos A sin B sin(A - B) = sin A cos B - cos A sin B
** See other side for more identities **
USEFUL TRIGONOMETRIC IDENTITIES
cos( - x) = - cos(x) cos( + x) = - cos(x) cos(2 - x) = cos(x) cos(2 + x) = cos(x)
Unit circle properties
sin( - x) = sin(x) sin( + x) = - sin(x) sin(2 - x) = - sin(x) sin(2 + x) = sin(x)
tan( - x) = - tan(x) tan( + x) = tan(x) tan(2 - x) = - tan(x) tan(2 + x) = tan(x)
Right-angled triangle properties
cos
2
-
x
= sin(x)
sin
2
-
x
= cos(x)
cos(x) = cos(x)
cos(x
+
2
)
=
-
sin(x)
cos(x + ) = - cos(x)
cos(x
+
3 2
)
=
sin(x)
cos(x + 2) = cos(x)
Shifting
by
2
cos(x) = cos(x)
cos(x
-
2
)
=
sin(x)
cos(x - ) = - cos(x)
cos(x
-
3 2
)
=
-
sin(x)
cos(x - 2) = cos(x)
cos(-x) = cos(x)
cos(
2
-
x)
=
sin(x)
cos( - x) = - cos(x)
cos(
3 2
-
x)
=
-
sin(x)
cos(2 - x) = cos(x)
sin(x) = sin(x)
sin(x
+
2
)
=
cos(x)
sin(x + ) = - sin(x)
sin(x
+
3 2
)
=
-
cos(x)
sin(x + 2) = sin(x)
sin(x) = sin(x)
sin(x
-
2
)
=
-
cos(x)
sin(x - ) = - sin(x)
sin(x
-
3 2
)
=
cos(x)
sin(x - 2) = sin(x)
sin(-x) = - sin(x)
sin(
2
-
x)
=
cos(x)
sin( - x) = sin(x)
sin(
3 2
-
x)
=
-
cos(x)
sin(2 - x) = - sin(x)
tan(x) = tan(x)
tan(x
+
2
)
=
-
cot(x)
tan(x + ) = tan(x)
tan(x
+
3 2
)
=
-
cot(x)
tan(x + 2) = tan(x)
tan(x) = tan(x)
tan(x
-
2
)
=
-
cot(x)
tan(x - ) = tan(x)
tan(x
-
3 2
)
=
-
cot(x)
tan(x - 2) = tan(x)
tan(-x) = - tan(x)
tan(
2
-
x)
=
cot(x)
tan( - x) = - tan(x)
tan(
3 2
-
x)
=
cot(x)
tan(2 - x) = - tan(x)
** See other side for more identities **
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