# Built-in Trigonometric functions in R

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.