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Oriented Angles and the Unit Circle

Angles in Euclidean Geometry and Trigonometry

In Euclidean geometry, an angle is the portion of the plane between two rays which have a common endpoint, and its measure can be any value between 0° and 360°. In Trigonometry, when representing angles in the unit circle, which is the circle with center at and radius , we usually add the concept of orientation of an angle, that allows us to define angles with any measure, even outside the Euclidean interval . In this activity we will refer to angles in the unit circle. An angle in the unit circle is in standard position when its vertex is and the initial side lies along the positive x-axis.

Sign of an Angle, Coterminal Angles and Primary Directed Angle

If we call one of the sides of the angle the initial side, and the other one terminal side, we have an oriented (or directed) angle. This representation allows us to give a sign to the angle: when measuring the angle counterclockwise, the sign of the angle is positive, otherwise it will be negative. When two angles in standard position have coincident terminal sides, they are called coterminal angles, such as and , or and . Any time that the measure of an angle is less than or greater than , we can associate it with its primary directed angle , which is coterminal with and whose measure is between and . This means that , with .


Use the slider or enter an angle measure in the input box (without the degree symbol) to explore coterminal and primary directed angles.