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Difficulty: HardRadians and Degrees

In the xyxy-plane, an angle θ\theta is in standard position. The terminal ray of θ\theta is rotated counterclockwise by 7π6\frac{7\pi}{6} radians, and then rotated clockwise by 135∘135^\circ. If the terminal ray of the resulting angle lies on the line y=−xy = -x in the fourth quadrant, and the original angle θ\theta has a measure of dd degrees, where 0≤d<3600 \leq d < 360, what is the value of dd?

Answer: 240

Answer

The value of dd is 240.
A terminal ray in the fourth quadrant lying on the line y=−xy = -x forms a 315∘315^\circ angle in standard position. The first rotation is counterclockwise by 7π6\frac{7\pi}{6} radians, which is equivalent to 7π6×180∘π=210∘\frac{7\pi}{6} \times \frac{180^\circ}{\pi} = 210^\circ. The second rotation is clockwise by 135∘135^\circ, representing a decrease of 135∘135^\circ. Therefore, the total transformation is θ+210∘−135∘=315∘\theta + 210^\circ - 135^\circ = 315^\circ. Solving for θ\theta gives θ+75∘=315∘\theta + 75^\circ = 315^\circ, which simplifies to θ=240∘\theta = 240^\circ. Since 240∘240^\circ lies in the interval [0,360)[0, 360), the value of dd is 240.

Step-by-Step Solution

1
Determine the angle in standard position for a terminal ray on the line y=−xy = -x in the fourth quadrant.
The terminal ray corresponds to an angle of 315∘315^\circ (or any angle coterminal with it).
The line y=−xy = -x in the fourth quadrant makes an angle of 45∘45^\circ below the positive xx-axis, which corresponds to 360∘−45∘=315∘360^\circ - 45^\circ = 315^\circ in standard position.
2
Convert the counterclockwise rotation from radians to degrees.
7π6 radians=210∘\frac{7\pi}{6} \text{ radians} = 210^\circ.
To convert radians to degrees, multiply the angle in radians by 180∘π\frac{180^\circ}{\pi}.
3
Express the rotations mathematically and set up the equation for θ\theta.
θ+210∘−135∘=315∘+360∘n\theta + 210^\circ - 135^\circ = 315^\circ + 360^\circ n (where nn is an integer).
In standard position, counterclockwise rotations represent positive changes in angle measure, whereas clockwise rotations represent negative changes in angle measure.
4
Solve for θ\theta and apply the domain restriction 0≤d<3600 \leq d < 360.
θ=240∘\theta = 240^\circ, so d=240d = 240.
Simplifying the equation gives θ+75∘=315∘\theta + 75^\circ = 315^\circ, which yields θ=240∘\theta = 240^\circ when n=0n=0.

Key Concept

Converting angles from radians to degrees, understanding the direction of rotation, and determining standard position angles on the coordinate plane.
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Topics

Radians and Degrees

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