Scheme Programming/Further Maths

Trigonometric FunctionsEdit

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 FunctionsEdit

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 FunctionsEdit

Raising a base to a powerEdit

Scheme provides the `expt` function to raise a base to an exponent.

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

Finding the square root of a numberEdit

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 functionsEdit

ExponentialEdit

Scheme provides a `exp` function for raising base ${\displaystyle e}$  to a power:

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

LogarithmEdit

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 ${\displaystyle e}$ . Instead, you can type

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

Other useful maths functions (rounding, modulo, gcd, etc.)Edit

Rounding functionsEdit

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 divisionEdit

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
> (remainder x y)
error
> (remainder 2 1)
0
```
• `(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.