Positive and Negative Angles on a Unit Circle

The unit circle is a platform for describing all the possible angle measures from 0 to 360 degrees, all the negatives of those angles, plus all the multiples of the positive and negative angles from negative infinity to positive infinity. In other words, the unit circle shows you all the angles that exist.

Because a right triangle can only measure angles of 90 degrees or less, the circle allows for a much-broader range.

Positive angles

The positive angles on the unit circle are measured with the initial side on the positive x-axis and the terminal side moving counterclockwise around the origin. The figure shows some positive angles labeled in both degrees and radians.

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Notice that the terminal sides of the angles measuring 30 degrees and 210 degrees, 60 degrees and 240 degrees, and so on form straight lines. This fact is to be expected because the angles are 180 degrees apart, and a straight angle measures 180 degrees. You see the significance of this fact when you deal with the trig functions for these angles.

Negative angles

Just when you thought that angles measuring up to 360 degrees or 2π radians was enough for anyone, you’re confronted with the reality that many of the basic angles have negative values and even multiples of themselves. If you measure angles clockwise instead of counterclockwise, then the angles have negative measures:

A 30-degree angle is the same as an angle measuring –330 degrees, because they have the same terminal side. Likewise, an angle of

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is the same as an angle of

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But wait — you have even more ways to name an angle. By doing a complete rotation of two (or more) and adding or subtracting 360 degrees or a multiple of it before settling on the angle’s terminal side, you can get an infinite number of angle measures, both positive and negative, for the same basic angle.

For example, an angle of 60 degrees has the same terminal side as that of a 420-degree angle and a –300-degree angle. The figure shows many names for the same 60-degree angle in both degrees and radians.

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Although this name-calling of angles may seem pointless at first, there’s more to it than arbitrarily using negatives or multiples of angles just to be difficult. The angles that are related to one another have trig functions that are also related, if not the same.

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