# SINE / COSINE / TANGENT

In order to define the trigonometric functions for the angle, start with a right triangle definition.

Right triangle always includes a 90° (π/2 radians) angle. SINE of an angel  is a trigonometric function defines as length of opposite side divided by the length of Hypotenuse (longest side of the right triangle, opposite of right angel).

COSINE of an angel is a trigonometric function defines as length of adjacent side divided by the length of Hypotenuse (longest side of the right triangle, opposite of right angel).

TANGENT of angle is a trigonometric function defines as length of opposite side divided by the length of adjacent side.

We have three more trigonometric functions which are rarely used:

COTANGENT, SECANT and COSECANT

Cotangent can be derived in two ways:

cot x = 1 / tan x

or

cot x = cos x / sin x

SECANT of an angel is a trigonometric function defines as length of Hypotenuse  divided by the length of adjacent side.

sec x = 1 / cos x

COSECANT :

csc x = 1 / sin x

A radian is a n angle made at the center of circle by an arc which ai equal to the length I=of the readies of the particular circle, It is therefore a unit that is used to ensure an angel. ' ## Sines and cosines of special angles The Cosine Function is up-down curve (which repeats every 2π radians, or 360°) which starts at 1, heads up to 0 by π/2 radians (90°) and then heads down to −1. The Sine Function is up-down curve (which repeats every 2π radians, or 360°) which starts at 0, heads up to 1 by π/2 radians (90°) and then heads down to −1. Sine and cosine functions looks totally the same with shift for π/2 radians. The Tangent function has a different shape. it goes between negative and positive Infinity, crossing through 0, and at every π radians (180°), as shown on this plot. At π/2 radians (90°), and at −π/2 (−90°), 3π/2 (270°), etc, the function is officially undefined, because it could be positive Infinity or negative Infinity '

### Graphs of the Sine and Cosine Function

Graph of y=sin(x) function you can see: Notice how the sine values are positive between 0 and π, which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between π and 2π, which correspond to the values of the sine function in quadrants III and IV on the unit circle. Graph for y=cos(x) function: In both graphs the shape of the graph repeats after 2π. It means the functions are periodic with a period of 2π.

A periodic function is a function for which a specific horizontal shift, P, results in a function equal to the original function: f (x + P) = f (x) for all values of x in the domain of f. When this occurs, we call the smallest such horizontal shift with P > 0 the period of the function '

As you see cosine is a symmetric function.

### Sinusoidal Functions

Sine and cosine functions have a regular period and range. If we watch ocean waves or ripples on a pond, we will see that they resemble the sine or cosine functions. However, they are not necessarily identical. Some are taller or longer than others. A function that has the same general shape as a sine or cosine function is known as a sinusoidal function. The general forms of sinusoidal functions are

```y = A sin (Bx − C) + D
and

y = A cos (Bx − C) + D
```

f we let C = 0 and D = 0 in the general form equations of the sine and cosine functions, we obtain the forms

```y = A sin (Bx)
y = A cos (Bx)
``` If we choose different A we will se different graphs: In general form C is the reason of horizontal shift of graph: And D is the reason of vertical shift of graph:  Based on graph we have we can identify key parameters for the function: y=3sin(x/2)+1