The unit circle is a circle of radius centered at the origin in the coordinate plane.
The unit circle is an essential tool for understanding trigonometry.
Suppose that we want to construct an angle in the coordinate plane. There is a standard way in which we do this.
First, we draw the unit circle.
Then, we represent as a central angle of the unit circle. The angle has its vertex at the origin, its initial side lies on the positive -axis, and it rotates such that a positive angle is measured counter-clockwise.
For example, we represent the angles and as follows:
Construct a unit circle in the coordinate plane with 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, its vertex at the origin, and it rotates such that a positive angle is measured counter-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 $260^\circ$ 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 $30^\circ$ 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, we 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, its vertex at the origin, and it rotates such that a positive angle is measured counter-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 $\dfrac{7\pi}{6}$ 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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The unit circle turns out to be a really useful tool to help us to understand the trigonometric functions. However, sometimes, we'd like to represent an angle in the coordinate plane without reference to the unit circle.
Thankfully, the standard way that we represent angles in the coordinate plane without reference to the unit circle is pretty much the same as with the unit circle!
For example, the angles and can be represented in the coordinate plane as follows:
How is the angle normally represented in the coordinate plane?
First, we recall that:
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.
Therefore, the correct diagram is as follows:
How is the angle $\dfrac \pi 6$ normally represented in the coordinate plane?
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How is the angle $300^\circ$ normally represented in the coordinate plane?
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Given that the point has coordinates calculate the angle
First, we can form a triangle by creating a vertical line from to the -axis.
Using the tangent ratio, we have
Finally, using the inverse tangent, we get
Given that the point $A$ has coordinates $(-3, \sqrt 3)$, calculate the angle $\theta.$
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$240^\circ$ |
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$120^\circ$ |
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$135^\circ$ |
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$150^\circ$ |
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$300^\circ$ |
Given that the point $A$ has coordinates $(1,\sqrt 3),$ calculate the angle $\theta.$
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$60^\circ$ |
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$50^\circ$ |
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$62^\circ$ |
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$55^\circ$ |
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$65^\circ$ |
