Tables of trigonometric functions

Properties of trigonometric functions

Representation by other trigonometric functions

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