Some trigonometric identities - Ohio State University

Some trigonometric identities

Periodicity: sin x, cos x have period 2; tan x, cot x have period

Symmetry: cos is an even function, sin, tan, cot are odd.

Turning sin to cos:

sin(

2

-

x)

=

cos

x

cos(

2

-

x)

=

sin x

Relating tan and sec: 1 + tan2 x = sec2 x , 1 + cot2 x = csc2 x

Expanding sums:

sin(x ? y) = sin x cos y ? cos x sin y , cos(x ? y) = cos x cos y sin x sin y

Products to sums:

1 x+y

x-y

sin x cos y = sin

+ sin

2

2

2

1 y+x

y-x

1 y+x

y-x

cos x cos y = cos

+ cos

, sin x sin y = cos

- cos

2

2

2

2

2

2

Expand double angle:

sin(2x) = 2 sin x cos x , cos(2x) = cos2 x - sin2 x = 2 cos2 x - 1 = 1 - 2 sin2 x

or the other way around, go to double angle by

1 sin x cos = sin(2x) ,

cos2 x = 1 + cos(2x)

,

sin2 x = 1 - cos(2x)

2

2

2

Some integrals:

sec x dx = ln | sec x + tan x| + C , csc x dx = - ln | csc x + cot x| + C

Derivatives of inverse trigonometric functions:

d

1

d

1

d

1

arcsin x =

dx

1 - x2

,

arccos x = -

dx

1 - x2

,

dx arctan x = 1 + x2

Hyperbolic functions

Definitions: Relations:

ex - e-x

ex + e-x

sinh x =

, cosh x =

2

2

cosh2 x - sinh2 x = 1 ,

d sinh x = cosh x ,

d cosh x = sinh x

dx

dx

sinh-1 x = ln x + 1 + x2

, d sinh-1 x = 1

dx

1 + x2

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