Trigonometric Functions of Any Angle

The definitions of the trigonometric functions can be expanded to incorporate (or measure) any angle. Instead of adhering to the angular constraints of a right triangle, where an angle θ must be acute so that 0° < θ < 90°, these functions can be reapplied to any other angle θ that sits on a coordinate plane in standard position with the point (x,y) on its terminal side.

 

The length of the terminal side r can be found by using the following equation derived from the Pythagorean Theorem,

 

using x and y from the coordinate point as substitutes.

The definitions of the trigonometric functions of any angle are as follows:

 

 

In order to determine the sign of the trigonometric functions, one must determine what quadrant they are in.

For example, since sin θ = y/r, that means that sin θ is positive whenever y > 0, which is in quadrants I and II (r is always positive).

Reference Angles

Suppose θ is an angle in standard position. Its reference angle is the acute angle θ' (read as "theta prime") formed by the terminal side of  angle θ and the x-axis (horizontal axis). Reference angles are calculated based on what quadrant the terminal side of θ lies in.

 

Quadrant I:

 (radians and degrees)

 

 

Quadrant II:

 (radians)

 (degrees)

 

 

Quadrant III:

 (radians)

 (degrees)

 

 

Quadrant IV:

 (radians)

 (degrees)

 

Reference angles can also be used to determine the value of a trigonometric function. For instance, suppose one must find cos 480°. One can subtract 360° from 480° to get a co-terminal angle of 120°. The reference angle of 120° (which, in this case, is θ) is 60° because 120° is in Quadrant II, so to find θ' one simply subtracts 120° from 180° to get 60°. From there, one can deduce that cos 60° is negative because it is in Quadrant II (where x < 0, given that cos θ = x/r). Therefore, cos 480° can be rewritten as -cos 60° which is -1/2.

 



In other words, to evaluate a trigonometric function of any angle, one must calculate the reference angle and determine the function value of it. From there, one must figure out the appropriate sign of the function value by looking at what quadrant it is in.