Reference angle

What is a reference angle

Reference Angle: the acute angle between the terminal arm/terminal side and the x-axis. The reference angle is always positive.

In other words, the reference angle is an angle being sandwiched by the terminal side and the x-axis. It must be less than 90 degree, and always positive.

A good thing to note before we move on is that when you're on the positive x-axis, the angle is 0° or 360°, which is also known as $2 \pi$ radians. When you get to the positive y-axis, you'll get 90° or $\frac{\pi}{2}$ radians. The negative x-axis gets you to 180° or $\pi$ radians. Lastly, the negative y-axis brings you to 270° or $\frac{3 \pi}{2}$ radians. Knowing these, you can use them as shortcuts to help you determine around where a reference angle should be and if you're in the right range.

Let's try out a few example questions on how to find reference angles.

Question 1:

Reference Angle: the acute angle between the terminal arm and the x-axis; reference angle is always positive.

Determine the reference angle of 130°.

Solution:

We want to determine the reference angle of a 130 degree angle. First we draw the standard angle of 130 degrees on a xy plane. Starting from the x axis (zero), we turn the terminal arm to the positive direction since we are dealing with a positive angle. We stop at 130 degree and get the standard angle

From the graph, we know that the angle lands on the second quadrant ( or quadrant 2 ). Now, we can determine the reference angle.

Based on the definition of a reference angle, we can determine that the reference angle is 50 degree

Question 2:

Determine the reference angle of 200°.

Solution:

In this question, we are looking for the reference angle of 200 degree. Same as the last example, we draw the standard angle of 200 degrees on a xy plane. Starting from the x axis (zero), we turn the terminal arm to the positive direction. We stop turning the terminal arm when it reaches 200 degree, and we get the standard angle.

Now, according to the definition of a reference angle, we can determine that the reference angle of 200 degree is 20 degree. And we know that the standard and reference angle land on the third quadrant ( or quadrant 3 )

Question 3:

Determine the reference angle of -23°.

Solution:

This time, we are going to find the reference angle of a negative angle: -23 degree. Same as the last example, we draw the standard angle of -23 degree on a xy plane. Also, starting from the x axis (zero), however, this time we turn the terminal arm to the negative direction. At the end, we know that the standard angle = -23 degree.

Now, according to the definition of reference angle, we can determine that the reference angle of -23 degree is 23 degree. Remember, the reference angle is always positive.

Try playing around with this online calculator to help you determine the reference angles of a certain degree. It'll help you check your answer!

You can revise trigonometric rations in radians to help you revise radian measures. As a refresher, it's just another way to measure angles other than degrees. Do also revise coterminal angles, which are also angles in standard positions with a common terminal side.