Trigonometric Functions

Definition of Trigonometric Functions

In the coordinate plane, let the center be at the origin and a point on a circle with radius (where ).

Let the positive direction of the -axis be the initial side. When the angle represented by the terminal side is , the trigonometric functions for are defined as follows:

^[1], ^[2], ^[3]

Here, , , and are called the sine function, cosine function, and tangent function, respectively. These are referred to as the trigonometric functions of .

1. The signs of trigonometric functions in each quadrant are as follows: ^[4]

2. Quadrant	1st Quadrant	2nd Quadrant	3rd Quadrant	4th Quadrant
   	[4:1]	[4:2]	[4:3]	[4:4]
   	[4:5]	[4:6]	[4:7]	[4:8]
   	[4:9]	[4:10]	[4:11]	[4:12]

3. is not defined at (where is an integer).

Relationships Between Trigonometric Functions

If the angle represents the terminal side that intersects the unit circle at the point , the following hold true:

1. Since , , and (where ), we have:
   
2. Since point lies on the circle , the following equation holds:
   

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1. ↩︎
2. ↩︎
3. ↩︎

4. Quadrant	1st Quadrant	2nd Quadrant	3rd Quadrant	4th Quadrant

   ↩︎ ↩︎ ↩︎ ↩︎ ↩︎ ↩︎ ↩︎ ↩︎ ↩︎ ↩︎ ↩︎ ↩︎ ↩︎