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Difficulty: MediumRadians and Degrees

In the xyxy-plane, an angle θ\theta is in standard position. The terminal ray of the angle is rotated counterclockwise by 7π6\frac{7\pi}{6} radians, and then rotated clockwise by 45∘45^\circ. If the terminal ray of the resulting angle lies on the positive yy-axis, which of the following could be the value of θ\theta, in degrees?

  1. A
    15∘15^\circ
  2. B
    195∘195^\circ
  3. C
    255∘255^\circ
  4. 285∘285^\circAnswer

Answer

The value of θ\theta could be 285∘285^\circ.
To find the value of θ\theta, we first convert the counterclockwise rotation of 7π6\frac{7\pi}{6} radians into degrees. Multiplying 7π6\frac{7\pi}{6} by 180π\frac{180}{\pi} yields 210∘210^\circ. A counterclockwise rotation increases the angle measure, so we add 210∘210^\circ. A clockwise rotation decreases the angle measure, so we subtract 45∘45^\circ. The resulting angle is θ+210∘−45∘=θ+165∘\theta + 210^\circ - 45^\circ = \theta + 165^\circ. Since the terminal ray of the resulting angle lies on the positive yy-axis, it is coterminal with 90∘90^\circ. Setting θ+165∘=90∘+360∘k\theta + 165^\circ = 90^\circ + 360^\circ k for an integer kk gives θ=−75∘+360∘k\theta = -75^\circ + 360^\circ k. For k=1k = 1, we get θ=285∘\theta = 285^\circ, which is the correct value.

Step-by-Step Solution

1
Convert the counterclockwise rotation from radians to degrees.
7π6×180π=210∘\frac{7\pi}{6} \times \frac{180}{\pi} = 210^\circ
To work with degrees consistently, we convert the radian measure of the rotation using the conversion factor 180π\frac{180}{\pi}.
2
Set up the equation for the final angle position using the directions of rotation.
Final angle = θ+210∘−45∘=θ+165∘\theta + 210^\circ - 45^\circ = \theta + 165^\circ
Counterclockwise rotations are positive (added to θ\theta) and clockwise rotations are negative (subtracted from the result).
3
Relate the final angle to the positive yy-axis and solve for θ\theta.
θ+165∘=90∘+360∘k  ⟹  θ=−75∘+360∘k\theta + 165^\circ = 90^\circ + 360^\circ k \implies \theta = -75^\circ + 360^\circ k. For k=1k = 1, θ=285∘\theta = 285^\circ.
The positive yy-axis corresponds to 90∘90^\circ in standard position. Solving for θ\theta and adding multiples of 360∘360^\circ yields the possible measures of the starting angle.

Key Concept

Converting angle measures between radians and degrees, determining the direction of angular rotation, and finding coterminal angles.
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Topics

Radians and Degrees

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