# Trigonometric Functions

This is a trigonometric functions calculator that accepts real and complex numbers.
input pi for pi and sqrt(2) for square root of 2 for example.

This online tool calculates trigonometric functions : sine, cosine, tangent, cotangent, secant, and cosecant for a given angle in degrees or radians. Example: cos (2*pi/3).

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))