Definition of Trigonometric Functions

Trigonometric functions are defined with the help of coordinates of rotating point. Consider on x-y plane a circle with graphical definition of trigonometric functionsradius 1 and centered at origin (it is also called unit circle). Let's denote by `P_0` a point on a circle with coordinate (1;0); this point is called starting point. Let's take arbitrary number t and rotate starting point on angle t about point O (origin); if t>0 then we rotate counter-clockwise, if t<0 then we rotate clockwise.

As result of rotation we obtain on a circle point `P_t`.

Its y-coordinate is called sine of number t and denoted by `sin(t)`, and x-coordinate is called cosine of number t and denoted by `cos(t)`.

Tangent of number t is ratio of sine and cosine: `tan(t)=(sin(t))/(cos(t))`.

Cotangent of number t is ratio of cosine and sine: `cot(t)=(cos(t))/(sin(t))`.

Below is the table of values of sine, cosine, tangent and cotangent for some angles:

Function Argument t
`0^0` `30^0` `45^0` `60^0` `90^0` `180^0` `270^0`
sin(t) 0 `1/2` `(sqrt(2))/2` `(sqrt(3))/2` 1 0 -1
cos(t) 1 `(sqrt(3))/2` `(sqrt(2))/2` `1/2` 0 -1 0
tan(t) 0 `(sqrt(3))/3` 1 `sqrt(3)` - 0 -
cot(t) - `sqrt(3)` 1 `(sqrt(3))/3` 0 - 0

From definitions it follows that there don't exist tangent of angles, whose cosine equals 0, and cotangent of angles, whose sine equals 0.

When we talk about sine, cosine, tangent and cotangent of number, we use both degree and radian measure of angle: `1\ rad=(180^0)/pi~~57^0`; `1^0=(pi)/(180)\ rad~~0.017\ rad`.

For example:

  1. `sin(4)~~sin(4*57^0)=sin(228^0)`;
  2. `cos(225^0)=cos(225*pi/(180))=cos((5pi)/4)`.

Functions `y=sin(x), y=cos(x),y=tan(x),y=cot(x)` are called trigonometric functions.

Sometimes, mathematicians use two additional trigonometric functions:

  1. Secant function: `sec(x)=1/(cos(x))`.
  2. Cosecant function: `csc(x)=1/(sin(x))`.