Examkin

Soru

Zorluk: ZorRadians 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?

Cevap: 240

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Konular

Radians and Degrees

Bu konudan daha fazla soru