
What is a Reference Angle?

The reference angle is the positive acute angle that can represent an angle of any measure. Basically, any angle on the xy plane has a reference angle, which is always between 0 and 90 degrees. The reference angle is always the smallest angle that you can make from the terminal side of an angle (ie where the angle ends) with the xaxis. A reference angle always uses the xaxis as its frame of reference.
Rules of Angles and Reference angle
Positive angles go in a counter clockwise direction. Below is a picture of a positive fifty degree angle
Quadrant I
Evrey positive angle in quadrant I is already acute...so the reference angle is the measure of the angle itself:
Quadrant II
To find the reference angle measuring x ° for angle in Quadrant III, the formula is $$ 180  x^{\circ} $$ .
Quadrant III
To find the reference angle measuring x ° for angle in Quadrant III, the formula is $$ x  180 ^{\circ} $$ .
Quadrant IV
To find the reference angle measuring x ° for angle in Quadrant IV, the formula is $$360 ^{\circ} x $$ .
What is the reference angle for the angle in the graph below?
Remember that the reference angle always uses the xaxis as a frame of reference.
What is the reference angle for a 210° angle?
Remember that the reference angle always uses the xaxis as a frame of reference.
What is the reference angle for a 300° angle?
Remember that the reference angle always uses the xaxis as a frame of reference.
Word Problems
Problem 1) What is the reference angle for an angle that measures 91 ° ?
For a quadratnt 2 angle, the reference angle is always 180°  given angle
In this case, $$ 180  91 = \color{Red}{89} $$

Problem 2) What is the reference angle for an angle that measures 250 ° ?
For a quadratnt 3 angle, the reference angle is always given angle  180
In this case, $$ 250  180= \color{Red}{ 70 } $$

