Finding the Reference Angle

By Reed Denney

Explanation:

The reference angle of an angle θ in standard position

is the acute angle r

formed by the terminal side of θ

and the horizontal axis.

When determining the reference angle vertically drop (or raise),

the line to the horizontal axis (x-axis). 

The degree of the acute angle,

with vertex on the origin,

in the triangle that is created

is the reference angle.

Calculating:

If the angle stays within the 1^st quadrant,

the degree of that angle is always the reference angle. 

If the angle expands to the 2^nd quadrant,

you would use the expression 180-the angle° =  the reference angle. 

If the angle expands to the 3^rd quadrant,

you would us the expression:

the angles°-180= the reference angle. 

If the angle expands all the way to the 4^th quadrant

you would use the expression: 360-the angles°= the reference angle.

Quadrantal Angles: Angles 0°, 90°, 180°, 270°, and 360° do not have reference angles because they are quadrantal angles.

Examples:

In the example above you have an angle of 135°.

If  you drop an imaginary line vertically down,  

you can see that it makes an angle of 45° which is the reference angle.

In this second example you have an angle of 240°.

  This time, since the angle is below the x-axis,

you would raise an imaginary line up to the x-axis. 

Thus creating an angle of 60°,

which is the reference angle for 240°.