Trigonometric functions are the part of built-in mathematical functions. All these functions are vectorised. In this tutorial you will learn some built-in trigonometric functions in R and how to use these trigonometric functions in R.

## Built-in Trigonometric functions

Commonly used trigonometric functions in R are listed below:

Function | Operation Performed |
---|---|

`sin(x)` |
sine value of `x` |

`cos(x)` |
cosine value of `x` |

`tan(x)` |
tangent value of `x` |

`asin(x)` |
arc-sine value of `x` |

`acos(x)` |
arc-cosine value of `x` |

`atan(x)` |
arc-tangent value of `x` |

`sinh(x)` |
hyperbolic sine of `x` |

`cosh(x)` |
hyperbolic cosine of `x` |

`tanh(x)` |
hyperbolic tangent of `x` |

`asinh(x)` |
hyperbolic arc-sine of `x` |

`acosh(x)` |
hyperbolic arc-cosine of `x` |

`atanh(x)` |
hyperbolic arc-tangent of `x` |

### Examples of Trigonometric functions

R uses the argument to the trigonometric functions `sin()`

, `cos()`

and `tan()`

in radians.

`sin()`

function in R

To compute the $\sin(30^o)$ in R, we need to convert degree to radian as $30^o = \frac{30*\pi}{180}$ radian.

```
# compute sine of 30 degree angle
sin(30*pi/180)
```

`[1] 0.5`

Note that `pi`

is Special constant in R.

`cos()`

function in R

To compute the $\cos(45^o)$ in R, we need to convert degree to radian as $45^o = \frac{45*\pi}{180}$ radian.

```
# compute cos of 45 degree angle
cos(45*pi/180)
```

`[1] 0.7071068`

`tan()`

function in R

To compute the $\tan(60^o)$ in R, we need to convert degree to radian as $60^o = \frac{60*\pi}{180}$ radian.

```
# Compute tan of 60 degree angle
tan(60*pi/180)
```

`[1] 1.732051`

R always works with angles in radian and not degrees. So remember to convert the angle from degree to radian while calculating trigonometric functions.

### Examples of Inverse Trigonometric functions

The inverse function returns the angle in radian. To convert it into degree, multiply the answer by $180/\pi$.

`asin()`

function in R

```
# Compute sin inverse of 0.5.
asin(0.5)*180/pi
```

`[1] 30`

`acos()`

function in R

```
# Compute cos inverse of 0.7071068.
acos(0.7071068)*180/pi
```

`[1] 45`

```
# Compute tan inverse of 60 degree
atan(1.732051)*180/pi
```

`[1] 60`

The result of the inverse trigonometric function is in radians. To get it the result in degrees multiply the result by $180/\pi$.

### Examples of Hyperbolic Trigonometric functions

`sinh()`

function in R

The hyperbolic sine of $x$ is given by

` $$ \sinh (x)=\frac{e^x- e^{-x}}{2} $$ `

`sinh(30*pi/180)`

`[1] 0.5478535`

`cosh()`

function in R

The hyperbolic cosine of $x$ is given by

` $$ \cosh (x)=\frac{e^x+ e^{-x}}{2} $$ `

`cosh(45*pi/180)`

`[1] 1.324609`

`tanh()`

function in R

The hyperbolic tangent of $x$ is given by

` $$ \tanh (x)=\frac{e^x- e^{-x}}{e^x+ e^{-x}} $$ `

`tanh(60*pi/180)`

`[1] 0.7807144`

### Examples of Inverse Hyperbolic Trigonometric functions

`asinh()`

function in R

The inverse hyperbolic sine of $x$ is given by

` $$ \text{asinh} (x) =\ln (x+\sqrt{x^2+1}) $$ `

To compute the inverse of sine hyperbolic of `0.5478535`

, use the following function. Multiplying the result by $180/\pi$ gives you angle in degree.

`asinh(0.5478535)*180/pi`

`[1] 30`

`acosh()`

function in R

The inverse hyperbolic cosine of $x$ is given by

` $$ \text{acosh} (x)=\ln (x+\sqrt{x^2-1}), x\geq 1 $$ `

To compute the inverse of cosine hyperbolic of `1.324609`

, use the following function. Multiplying the result by $180/\pi$ gives you angle in degree.

`acosh(1.324609)*180/pi`

`[1] 44.99999`

`atanh()`

function in R

The inverse hyperbolic tangent of $x$ is given by

` $$ \text{atanh} (x)=\frac{1}{2}\ln \bigg(\frac{1+x}{1-x}\bigg), |x| < 1 $$ `

To compute the inverse of cosine hyperbolic of `2.509178`

, use the following function. Multiplying the result by $180/\pi$ gives you angle in degree.

`atanh(0.7807144)*180/pi`

`[1] 59.99999`

## Endnote

In this tutorial you learned about trigonometric functions in R and how to use these functions in R.

To learn more about other built-in functions and user-defined functions in R, please refer to the following tutorials:

- Built-in Mathematical functions in R
- Built-in Special Mathematical functions in R
- Built-in Statistical functions in R
- Built-in Character functions in R
- User-defined functions in R Part I
- User-defined functions in R Part II
- Functions in R

Hopefully you enjoyed learning this tutorial on trigonometric functions in R. Hope the content is more than sufficient to understand trigonometric functions in R.