Reference angle

Definition: The smallest angle that the terminal side of a given angle makes with the x-axis.
Try this: Adjust the angle below by dragging the orange point around the origin, and note the blue reference angle.

In the figure above, as you drag the orange point around the origin, you can see the blue reference angle being drawn. It is the angle between the terminal side and the x axis. As the point moves into each quadrant, note how the reference angle is always the smallest angle between the terminal side and the x axis.

It is always positive

Regardless of which quadrant we are in, the reference angle is always made positive. Drag the point clockwise to make negative angles, and note how the reference angle remains positive.

It is always <= 90°

As you can see from the figure above, the reference angle is always less than or equal to 90°, even for very large angles. Drag the point around the origin several times. Note how the reference angle always remain less than or equal to 90°, even for large angles.

Finding the reference angle

1. If necessary, first "unwind" the angle:   Keep subtracting 360 from it until it is lies between 0 and 360°. (For negative angles add 360 instead).
2. Sketch the angle to see which quadrant it is in.
 2 1 3 4
3. Depending on the quadrant, find the reference angle:
 Quadrant Reference angle for θ 1 Same as θ 2 180 - θ 3 θ - 180 4 360 - θ

If you are working in radians, recall that 360° is equal to 2π radians, and 180° is equal to π radians.

What is it used for?

In trigonometry we use the functions of angles like sin, cos and tan. It turns out that angles that have the same reference angles always have the same trig function values (the sign may vary). So for example

sin(45) = 0.707

The angle 135° has a reference angle of 45°, so its sin will be the same. Checking on a calculator:

sin(135) = 0.707

This comes in handy because we only then need to memorize the trig function values of the angles less than 90°. The rest we can find by first finding the reference angle.

Also, when solving trigonometric equations we may notice one term,such as sin(x) and another, sin(π-x), and realize they are going to be equal, because the second is the reference angle of the first.

Things to try

1. In the figure above, click 'reset' and 'hide details'.
2. Drag the orange dot around the origin to a new location.
3. Calculate the reference angle for it