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Difficulty: HardUnit Circle and Angle Measures (Radians and Degrees)

An object on a unit circle starts at the point (1,0)(1,0) and rotates counterclockwise through an angle of 990∘990^\circ. It then rotates clockwise through an angle of 13π4\frac{13\pi}{4} radians. At which of the following coordinate points on the unit circle does the object finish its path?

  1. A
    (−22,22)\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)
  2. B
    (0,−1)(0, -1)
  3. (22,22)\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)Answer
  4. D
    (−1,0)(-1, 0)
  5. E
    (0,1)(0, 1)

Answer

The correct coordinate point on the unit circle is (22,22)\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right).
The correct coordinate point is found by converting the counterclockwise rotation of 990∘990^\circ to 11π2\frac{11\pi}{2} radians. Subtracting the clockwise rotation of 13π4\frac{13\pi}{4} radians yields 9π4\frac{9\pi}{4} radians. Simplifying this to its coterminal angle in the interval [0,2π)[0, 2\pi) gives π4\frac{\pi}{4} radians. The coordinates of π4\frac{\pi}{4} on the unit circle are (22,22)\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right).

Step-by-Step Solution

1
Convert the initial counterclockwise rotation from degrees to radians.
990∘×π180∘=11π2990^\circ \times \frac{\pi}{180^\circ} = \frac{11\pi}{2} radians.
To perform calculations with the second rotation which is given in radians, both angle measures should be in the same unit.
2
Subtract the clockwise rotation of 13π4\frac{13\pi}{4} radians from the first rotation.
11π2−13π4=22π4−13π4=9π4\frac{11\pi}{2} - \frac{13\pi}{4} = \frac{22\pi}{4} - \frac{13\pi}{4} = \frac{9\pi}{4} radians.
Clockwise rotation corresponds to subtracting the angle from the current position.
3
Find the coterminal angle of 9π4\frac{9\pi}{4} radians within the standard interval [0,2π)[0, 2\pi).
9π4−2π=π4\frac{9\pi}{4} - 2\pi = \frac{\pi}{4} radians.
Subtracting full rotations (2π2\pi radians) simplifies the angle to its principal equivalent on the unit circle.
4
Determine the coordinates on the unit circle corresponding to the angle π4\frac{\pi}{4}.
(cos⁡(π4),sin⁡(π4))=(22,22)\left(\cos\left(\frac{\pi}{4}\right), \sin\left(\frac{\pi}{4}\right)\right) = \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right).
On the unit circle, the coordinates of a point at angle θ\theta are (cos⁡(θ),sin⁡(θ))(\cos(\theta), \sin(\theta)).

Key Concept

Finding the position on the unit circle after multiple rotations by converting degrees to radians, calculating the coterminal angle, and evaluating coordinate values.
Estimated Time:2m 0s
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Topics

Unit Circle and Angle Measures (Radians and Degrees)

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