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.

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Some basic angles in radians:

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

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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

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As you see cosine is a symmetric function.

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