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How to Locate Reference Angles

Each of the angles positioned with its center at the origin has a reference angle, which is always a positive acute angle. By identifying the reference angle, you can determine the function values for that reference angle and, ultimately, the function values for the original angle.

Sometimes, solving for the reference angle first is much easier than trying to solve for the function of the original angle. The trig functions are periodic and so have values that repeat over and over; sometimes those values are positive, and sometimes they’re negative. Using a reference angle helps keep the number of different possibilities to a minimum. You just assign the positive or negative sign after determining which reference angle to match the original angle with.

You have many ways to name the same angle.
You have many ways to name the same angle.

You can determine a reference angle by looking at the terminal side of the angle you’re working with and the positive or negative x-axis, depending on which quadrant the terminal side is in:

  • Quadrant I (QI): The reference angle is the same as the original angle itself.

  • Quadrant II (QII): The reference angle is the measure from the terminal side down to the negative x-axis.

  • Quadrant III (QIII): The reference angle is the measure from the negative x-axis down to the terminal side.

  • Quadrant IV (QIV): The reference angle is the measure from the terminal side up to the positive x-axis.

The following figure shows the positions of the reference angles in the four quadrants.

You measure reference angles by using the <i>x</i>-axis.
You measure reference angles by using the x-axis.

As with all angles, you can measure reference angles in degrees or radians. You may prefer to work in degrees and then convert a radian measure to do these computations. Whichever method you choose is fine — it’s all a matter of taste.

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