Trigonometric Functions of Any Angle

In this section, the definitions of trigonometric functions are extended to cover any angle.

In trigonometry, we tend to think of an angle as created by a rotating ray. The beginning position of the ray is called the initial side. Usually, the initial side coincides with the positive x-axis; angles with such an initial side are said to be in standard position. The post-rotation position of the ray is called the terminal side. The measure of the angle is a number which describes the amount of rotation. If the rotation of the ray was clockwise, the angle measure is a negative number; if the rotation was counterclockwise, the angle measure is positive.


An angle θ in standard position creates a right triangle on the coordinate axes as follows:

1. Choose a point P0=(x0, y0) on the terminal side of the angle.
2. Draw a vertical line from P0 to the x-axis (note: this line is perpendicular to the x-axis).
3. The resulting right triangle with vertices (x0, y0), (x0, 0), (0,0) is called the reference triangle for the angle θ

The vertical side of this triangle has length y0; the horizontal side has length x0; the hypotenuse is the distance from P0 to the origin, or




Trigonometric Functions of any angle


















*Note: Notice that tan(θ) and sec(θ) are not defined when the terminal side of the angle is vertical; Also cot(θ) and csc(θ) are not defined with the terminal side of the angle is horizontal.





Evaluating Trigonometric Functions When Given a Point

To evaluate a trigonometric function just take the point given on the terminal side of the angle and plug it into the formula for the function you solving listed above in the table. You can think of r as the hypotenus of the reference triangle so