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

A terminal ray of an angle θ\theta in standard position passes through the point (−12,y)\left(-\frac{1}{2}, y\right) on the unit circle in Quadrant III. What is the value of θ\theta in radians, where 0≤θ<2π0 \le \theta < 2\pi?

  1. A
    2π3\frac{2\pi}{3}
  2. 4π3\frac{4\pi}{3}Answer
  3. C
    5π4\frac{5\pi}{4}
  4. D
    5π3\frac{5\pi}{3}
  5. E
    7π6\frac{7\pi}{6}

Answer

The correct answer is the option containing the radian measure 4π3\frac{4\pi}{3}.
The correct answer is the option containing the radian measure 4π3\frac{4\pi}{3}. Any point (x,y)(x, y) on the unit circle satisfies x2+y2=1x^2 + y^2 = 1. Substituting x=−12x = -\frac{1}{2} gives 14+y2=1\frac{1}{4} + y^2 = 1, which simplifies to y2=34y^2 = \frac{3}{4}, so y=±32y = \pm \frac{\sqrt{3}}{2}. Since the terminal ray lies in Quadrant III, the y-coordinate must be negative, so y=−32y = -\frac{\sqrt{3}}{2}. The angle θ\theta in [0,2π)[0, 2\pi) with cos⁡(θ)=−12\cos(\theta) = -\frac{1}{2} and sin⁡(θ)=−32\sin(\theta) = -\frac{\sqrt{3}}{2} is θ=4π3\theta = \frac{4\pi}{3} radians.

Step-by-Step Solution

1
Use the equation of the unit circle, x2+y2=1x^2 + y^2 = 1, to find the y-coordinate of the point.
Since x=−12x = -\frac{1}{2}, we have (−12)2+y2=1⇒14+y2=1⇒y2=34\left(-\frac{1}{2}\right)^2 + y^2 = 1 \Rightarrow \frac{1}{4} + y^2 = 1 \Rightarrow y^2 = \frac{3}{4}.
The coordinates of any point on the unit circle must satisfy the equation x2+y2=1x^2 + y^2 = 1.
2
Determine the correct sign of the y-coordinate using the quadrant information.
y=−32y = -\frac{\sqrt{3}}{2}
Since the point lies in Quadrant III, both the x and y coordinates must be negative.
3
Identify the angle θ\theta in the interval [0,2π)[0, 2\pi) that corresponds to these coordinates on the unit circle.
θ=4π3\theta = \frac{4\pi}{3} radians
On the unit circle, cos⁡(θ)=x=−12\cos(\theta) = x = -\frac{1}{2} and sin⁡(θ)=y=−32\sin(\theta) = y = -\frac{\sqrt{3}}{2}. In Quadrant III, the angle with a reference angle of π3\frac{\pi}{3} is π+π3=4π3\pi + \frac{\pi}{3} = \frac{4\pi}{3}.

Key Concept

Finding an angle in standard position on the unit circle given its x-coordinate and quadrant

Alternative Method

Alternatively, one can convert the options from radians to degrees and evaluate their cosine values. The correct angle must satisfy cos⁡(θ)=−1/2\cos(\theta) = -1/2. Since cos⁡(2π/3)=−1/2\cos(2\pi/3) = -1/2 and cos⁡(4π/3)=−1/2\cos(4\pi/3) = -1/2, we only need to check these two options. Among these, only 4π3\frac{4\pi}{3} lies in Quadrant III, which confirms it as the correct answer.
Estimated Time:1m 30s
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Topics

Unit Circle and Angle Measures (Radians and Degrees)

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