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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:
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:
  2. Quadrant 1st Quadrant 2nd Quadrant 3rd Quadrant 4th Quadrant
    Quadrant 1st Quadrant 2nd Quadrant 3rd Quadrant 4th Quadrant (x > 0,y > 0) (x < 0,y > 0) (x < 0,y < 0) (x > 0,y < 0) sin theta + + - - cos theta + - - + tan theta + - + -
  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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