# Scheme Programming/Further Maths

## Trigonometric Functions

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

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

### Raising a base to a power

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

> (expt 2 10)
1024


### Finding the square root of a number

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

### Exponential

Scheme provides a exp function for raising base $e$ to a power:

> (exp 2)
7.3890560989306504


### Logarithm

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 $e$. Instead, you can type

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


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

### Rounding functions

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

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.