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

A radar signal sweeps counterclockwise around a control tower located at the origin of a coordinate plane. Starting from standard position along the positive xx-axis, the radar line rotates through an angle of 13π4\frac{13\pi}{4} radians. Which of the following ordered pairs represents the (x,y)(x, y) coordinates of the point where the radar line intersects the unit circle centered at the origin?

  1. (−22,−22)\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)Answer
  2. B
    (−22,22)\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)
  3. C
    (22,−22)\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)
  4. D
    (−12,−32)\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)
  5. E
    (−32,−12)\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)

Answer

(−22,−22)\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)
Subtracting 2π2\pi (one full rotation) from 13π4\frac{13\pi}{4} gives the coterminal angle 5π4\frac{5\pi}{4}. This angle terminates in Quadrant III, where both cosine (xx-coordinate) and sine (yy-coordinate) are negative. Using the reference angle π4\frac{\pi}{4}, both values have magnitude 22\frac{\sqrt{2}}{2}, giving the coordinate pair (−22,−22)\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right).

Step-by-Step Solution

1
Find an equivalent coterminal angle within one full revolution [0,2π)[0, 2\pi).
Subtract 2π=8π42\pi = \frac{8\pi}{4} from 13π4\frac{13\pi}{4}: 13π4−8π4=5π4\frac{13\pi}{4} - \frac{8\pi}{4} = \frac{5\pi}{4} radians.
Coterminal angles share the exact same terminal ray and unit circle coordinates.
2
Identify the quadrant and reference angle for 5π4\frac{5\pi}{4}.
The angle lies in Quadrant III because π<5π4<3π2\pi < \frac{5\pi}{4} < \frac{3\pi}{2}. The reference angle is 5π4−π=π4\frac{5\pi}{4} - \pi = \frac{\pi}{4}.
The reference angle determines the absolute magnitude of the trigonometric coordinates.
3
Calculate the coordinates (x,y)=(cos⁡θ,sin⁡θ)(x, y) = (\cos\theta, \sin\theta) for the terminal ray.
Since cos⁡(π4)=22\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} and sin⁡(π4)=22\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}, and both coordinates are negative in Quadrant III, (x,y)=(−22,−22)(x, y) = \left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right).
Points on the unit circle are defined by (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta).

Key Concept

Coterminal Angles and Unit Circle Coordinates
Estimated Time:1m 0s
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Topics

Unit Circle and Angle Measures (Radians and Degrees)

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