Trigonometry (from Greek trigõnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged during the 3^{rd} century BC from applications of geometry to astronomical studies.

Trigonometry is most simply associated with planar right angle triangles (each of which is a two-dimensional triangle with one angle equal to 90 degrees). The applicability to non-right-angle triangles exists, but, since any non-right-angle triangle (on a flat plane) can be bisected to create two right-angle triangles, most problems can be reduced to calculations on right-angle triangles. Thus the majority of applications relate to right-angle triangles

What is angle

An angle which has its vertex at the origin and one side lying on the positive x-axis. It can have a measure which positive or negative and can be greater than 360°

If the direction of rotation is anticlockwise, angle is positive .If the direction of rotation is clockwise,angle is negative

Once you have made a full circle (360°) keep going and you will see that the angle is greater than 360° .In fact you can go around as many times as you like. The same thing happens when you go clockwise. The negative angle just keeps on increasing

It can be measured in degrees or radian

Degree and Radian

They both are unit of measurement of angles Radian: A unit of measure for angles. One radian is the angle made at the centre of a circle by an arc whose length is equal to the radius of the circle. Degree: If a rotation from the initial side to terminal side is (1/360) of a revolution, the angle is said to have a measure of one degree, written as
1°. A degree is divided into 60 minutes, and a minute is divided into 60 seconds . One sixtieth of a degree is called a minute, written as 1", and one sixtieth of a minute is called a second, written as 1'.
Thus, 1° = 60', 1' = 60"

Relation between Degree and Radian
2π radian = 360 °
π radian= 180 °
1 radian= (180/π) °

Degree

30°

45°

60°

90°

120°

180°

360°

Radian

π/6

π/4

π/3

π/2

2π/3

π

2π

Trigonometric Ratio's

In a right angle triangle ABC where B=90° ,we can define six ratio's for the two sides i.e Hypotenuse/Base, Base/Perpendicular,Perpendicular/Base,Base/hypotenuse,Hypotenuse/Perpendicular,Perpendicular/Hypotenuse
Trigonometric ratio's are defined as

$\sin \theta= \frac {Perpendicular}{Hypotenuse}$

$cosec \theta= \frac {Hypotenuse}{Perpendicular}$

$\cos \theta= \frac {Base}{Hypotenuse}$

$\sec \theta= \frac {Hypotenuse}{Base}$

$\tan \theta= \frac {Perpendicular}{Base}$

$\cot \theta= \frac {Base}{Perpendicular}$

Notice that each ratio in the right-hand column is the inverse, or the reciprocal, of the ratio in the left-hand column.
The reciprocal of sin θ is csc θ ; and vice-versa.
The reciprocal of cos θ is sec θ.
And the reciprocal of tan θ is cot θ
These are valid for acute angles.
We are now going to define them for any angles and they are called now the Trigonometric functions.