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.
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.
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.
|Quadrant||Reference angle for θ|
|1||Same as θ|
|2||180 - θ|
|3||θ - 180|
|4||360 - θ|
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) = 0.707The 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.