Recall that, in the coordinate plane, an angle in the standard position has its vertex at the origin, its initial side lies along the positive -axis, and it rotates such that a positive angle is measured counter-clockwise.
Since a positive angle is always measured counter-clockwise, this means that a negative angle is measured clockwise.
For example, we can represent the angles and as central angles of the unit circle (in the standard way) as follows:
Construct a unit circle in the coordinate plane that shows a central angle of in the standard position.
Recall that:
A unit circle has a radius of and is centered at the origin in the coordinate plane.
An angle in the standard position has its initial side on the positive -axis, and its vertex is at the origin.
It rotates such that a positive angle is measured counter-clockwise, and a negative angle is measured clockwise.
Therefore, the correct diagram is as follows:
Which of the following diagrams shows a unit circle in the coordinate plane with a central angle of $-160^\circ$ in the standard position?
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Which of the following diagrams shows a unit circle in the coordinate plane with a central angle of $-48^\circ$ in the standard position?
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Construct a unit circle in the coordinate plane with a central angle of in the standard position.
To help us to visualize the angle, we first convert the measure of the angle into degrees.
To convert the measure of an angle in radians to an equivalent measure in degrees, we multiply the measure in radians by This gives
Therefore, is equivalent to
Then, recall that:
A unit circle has a radius of and is centered at the origin in the coordinate plane.
An angle in the standard position has its initial side on the positive -axis and its vertex is at the origin.
A positive angle has a rotation that is measured counter-clockwise, and a negative angle is measured clockwise.
Therefore, the correct diagram is as follows:
Which of the following diagrams shows the unit circle in the coordinate plane with a central angle of $-\dfrac{5\pi}{8}$ in the standard position?
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Which of the following diagrams shows the unit circle in the coordinate plane with a central angle of $-\dfrac{\pi}{4}$ in the standard position?
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Sometimes, we wish to represent negative angles in the coordinate plane without reference to the unit circle.
As with positive angles, the convention for representing negative angles is the same as without the unit circle.
For example, we can represent the angles and in the coordinate plane in the standard way as follows:
Construct the angle in the coordinate plane in the usual way.
First, we recall that:
An angle in the standard position has its initial side on the positive -axis, and its vertex lies at the origin.
It rotates such that a positive angle is measured counter-clockwise, and a negative angle is measured clockwise.
Therefore, the correct diagram is as follows:
How is the angle $-\dfrac {5\pi}3$ normally represented in the coordinate plane?
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How is the angle $-120^\circ$ normally represented in the coordinate plane?
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