# Trigonometric Functions

input pi for `pi` and sqrt(2) for square root of 2 for example.

Trigonometric functions (also called circular functions or angle functions) can be defined with different approaches explained below :

- definitions using a right triangle

- definitions using the trigonometric circle

- definitions implying complex numbers

## Right triangle and trigonometric functions

Opposite side of angle Â: a

Adjacent side of angle Â: b

Hypotenuse of the angle Â: c

**Sinus**

sin Â = opposite side/hypotenuse = a/c

**Cosine**

cos Â = adjacent side/hypotenuse = b/c

**Tangent**

tan Â = opposite side/adjacent side = a/b

**Cotangent**

cot A = adjacent/opposite = b/a

**Secant**

sec A = hypotenuse/adjacent = c/b

Note that:

sec Â = 1/cos Â

**Cosecant**

csc A = hypotenuse/opposite = c/a

Note that:

csc Â = 1/sin Â

## Trigonometric Circle

We assume that an orthogonal coordinate system is defined by Ox and Oy axes.

The trigonometric circle is centered on point O and has radius 1. It is oriented counterclockwise i.e. from Ox to Oy.

Let M be a point of the circle and `alpha` a radians measurement of the angle (Ox, OM) then,

- the cosine of alpha, noted cos(`alpha`) is the horizontal coordinate of M.

- the sinus of alpha, noted sin (`alpha`) is the vertical coordinate of M.

## Trigonometric functions applied to complex numbers

Commonly used trigonometric functions can be extended to complex numbers as following,

We suppose that a and b are real numbers.

`sin(a + b*i) = sin(a)*cosh(b) + i cos(a)*sinh(b)`

`cos(a + b*i) = cos(a)*cosh(b) - i sin(a)*sinh(b)`

`tan(a + b*i) = (sin(2*a)+ i*sinh(2*b))/(cos(2*a)+cosh(2*b))`

## See also

Conversion of angle measurement

Inverse trigonometric functions