Tables of trigonometric functions

Properties of trigonometric functions

Representation by other trigonometric functions

The trigonometric functions sine, cosine and tangent can be expressed by other trigonometric functions. Here are some important identities:

sin2x

cos2x

tan2x

cot2x

sin2x

-

1-cos2x

tan2x1+tan2x

11+cot2x

cos2x

1-sin2x

-

11+tan2x

cot2x1+cot2x

tan2x

sin2x1-sin2x

1-cos2xcos2x

-

1cot2x

cot2x

1-sin2xsin2x

cos2x1-cos2x

1tan2x

-

Function values of trigonometric functions

Radian

0

π6

π4

π3

π2

Degree

30°

45°

60°

90°

sin2x

0

12

122

123

1

cos2x

1

123

122

12

0

tan2x

0

133

1

3

-

cot2x

-

3

1

133

0

Reduction formulas

in degrees:

sin(90°+x)=cos(x)

cos(90°+x)=-sin(x)

tan(90°+x)=-cot(x)

cot(90°+x)=-tan(x)

sin(180°+x)=-sin(x)

cos(180°+x)=-cos(x)

tan(180°+x)=tan(x)

cot(180°+x)=cot(x)

in radians:

sin(π2+x)=cos(x)

cos(π2+x)=-sin(x)

tan(π2+x)=-cot(x)

cot(π2+x)=-tan(x)

sin(π+x)=-sin(x)

cos(π+x)=-cos(x)

tan(π+x)=tan(x)

cot(π+x)=cot(x)

Context of the trigonometric functions for the same argument

sin(x)2+cos(x)2=1

tan(x)=sin(x)cos(x)

cot(x)=cos(x)sin(x)=1tan(x)

cot(x)tan(x)=1

Addition theorems

Addition theorems of trigonometric functions

sin(x±y)=sin(x)cos(y)±cos(x)sin(y)

cos(x±y)=cos(x)cos(y)sin(x)sin(y)

tan(x±y)=tan(x)±tan(y)1tan(x)tan(y)

and for multiples of the argument value

sin(2x)=2sin(x)cos(x)

sin(3x)=3sin(x)-4sin3(x)

cos(2x)=cos2(x)-sin2(x)

cos(3x)=4cos3(x)-3cos(x)

Half of the argument value

sin(x2)=±1-cos(x)2

cos(x2)=±1+cos(x)2

tan(x2)=±1-cos(x)1+cos(x)

Sum and difference of trigonometric functions

sin(x)+sin(y)=2sin(x+y2)cos(x-y2)

cos(x)+cos(y)=2cos(x+y2)cos(x-y2)

sin(x)-sin(y)=2cos(x+y2)sin(x-y2)

cos(x)-cos(y)=-2sin(x+y2)sin(x-y2)

cos(x)±sin(x)=2sin(π4±x)=2cos(π4x)

tan(x)±tan(y)=sin(x±y)cos(x)cos(y)

cot(x)±cot(y)=±sin(x±y)sin(x)sin(y)

tan(x)+cot(y)=cos(x-y)cos(x)sin(y)

cot(x)-tan(y)=cos(x+y)sin(x)cos(y)

Products of trigonometric functions

sin(x)sin(y)=12(cos(x-y)-cos(x+y))

cos(x)cos(y)=12(cos(x-y)+cos(x+y))

sin(x)cos(y)=12(sin(x-y)+sin(x+y))

cos(x)sin(y)=12(sin(x+y)-sin(x-y))

Potencies of trigonometric functions

sin2(x)=12(1-cos(2x))

sin3(x)=14(3sin(x)-sin(3x))

sinn(x)=12n k = 0 n ( n k ) cos((n-2k)(x-π2))

cos2(x)=12(1+cos(2x))

cos3(x)=14(3cos(x)+cos(3x))

cosn(x)=12n k = 0 n ( n k ) cos((n-2k)x)

Releated sites

Here is a list of of further useful sites:

Index Trigonometry Power and n-th argument Derivative sine cosine tangent Sine, Cosine, Tangent calculator Triangle Sine (sin) Plot Cosine (cos) Plot Beat frequencies Plot