USEFUL TRIGONOMETRIC IDENTITIES - The University of Adelaide

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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