# Scheme Programming/Further Maths

## Contents

## Trigonometric Functions[edit]

Scheme always uses radians for its internal representation of angles, so its sine, cosine, tangent, arcsine, arccosine, and arctangent functions operate as such:

```
> (sin 0)
0.0
> (cos 0)
1.0
> (tan 0)
0.0
> (asin 1)
1.5707963267948965
> (acos 0)
1.5707963267948965
> (atan 1)
0.7853981633974483
```

## Hyperbolic Functions[edit]

Scheme provides a number of hyperbolic functions, such as hyperbolic sine, cosine, tangent and their inverses.

```
> (sinh 0)
0.0
> (cosh 0)
1.0
> (tanh 1)
0.7615941559557649
> (asinh 0)
0.0
> (acosh 1)
0.0
> (atanh 0)
0.0
```

## Power Functions[edit]

### Raising a base to a power[edit]

Scheme provides the `expt`

function to raise a base to an exponent.

```
> (expt 2 10)
1024
```

### Finding the square root of a number[edit]

Scheme provides a `sqrt`

function for finding the square root of a number.

```
> (sqrt 2)
1.4142135623730951
> (expt 2 0.5)
1.4142135623730951
```

## Exponential and logarithmic functions[edit]

### Exponential[edit]

Scheme provides a `exp`

function for raising base to a power:

```
> (exp 2)
7.3890560989306504
```

### Logarithm[edit]

Scheme provides a `log`

function for finding the natural logarithm of a number:

```
> (log 7.389056)
1.999999986611192
```

Note that there is no built-in procedure for finding any other base logarithm other than base . Instead, you can type

```
> (define logB
(lambda (x B)
(/ (log x) (log B))))
```

## Other useful maths functions (rounding, modulo, gcd, etc.)[edit]

### Rounding functions[edit]

Scheme provides a set of functions for rounding a real number up, down or to the nearest integer:

`(floor x)`

- This returns the largest integer that is no larger than x.`(ceiling x)`

- This returns the smallest integer that is no smaller than x.`(truncate x)`

- This returns the integer value closest to x that is no larger than the absolute value of x.`(round x)`

- This rounds value of x to the nearest integer as is usual in mathematics. It even works when halfway between values.`(abs x)`

- This returns the absolute value of x.

### Number theoretic division[edit]

In order to perform mathematically exact divisions and accomplish tasks for number theorists, Scheme provides a small number of division specific functions:

`(remainder x y)`

- Calculates the remainder of dividing y into x (that is, the remainder of`x/y`

):

```
> (remainder 5 4)
1
> (remainder -5 4)
-1
> (remainder 5 -4)
1
> (remainder -5 -4)
-1
```

`(modulo x y)`

- Calculates the modulo of x and y.

```
> (modulo 5 4)
1
> (modulo -5 4)
3
> (modulo 5 -4)
-3
> (modulo -5 -4)
-1
```

There is clearly a difference between modulo and remainder, one of them not shown here is that remainder is the only one which will return an inexact value, and can take inexact arguments.