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Zorluk: Çok zorUnit Circle and Angle Measures (Radians and Degrees)

A particle starts at the point (1,0)(1, 0) on the unit circle in the standard coordinate plane. It first travels along the circle counterclockwise by 11π4\frac{11\pi}{4} radians, then travels clockwise by 120∘120^\circ, and finally travels counterclockwise by π4\frac{\pi}{4} radians. Which of the following ordered pairs represents the coordinates of the particle's final position?

  1. (12,32)\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)Cevap
  2. B
    (12,−32)\left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right)
  3. C
    (22,22)\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)
  4. D
    (1,0)(1, 0)
  5. E
    (32,−12)\left( \frac{\sqrt{3}}{2}, -\frac{1}{2} \right)

Cevap

(12,32)\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)
The correct answer is the coordinate pair representing a net rotation of π3\frac{\pi}{3} radians. Since the particle starts at (1,0)(1, 0), its coordinates after a net rotation of π3\frac{\pi}{3} radians are given by (cos⁡(π3),sin⁡(π3))(\cos(\frac{\pi}{3}), \sin(\frac{\pi}{3})), which evaluates to (12,32)\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right).

Adım Adım Çözüm

1
Determine the sign and radian value of each rotation.
The first rotation (counterclockwise) is +11π4+\frac{11\pi}{4} radians. The second rotation (clockwise) is negative: −120∘=−120∘×π180∘=−2π3-120^\circ = -120^\circ \times \frac{\pi}{180^\circ} = -\frac{2\pi}{3} radians. The third rotation (counterclockwise) is +π4+\frac{\pi}{4} radians.
Standard orientation defines counterclockwise rotation as positive and clockwise rotation as negative. Angles must be in the same unit (radians) to be combined.
2
Calculate the net angle of rotation by summing the signed values.
θnet=11π4−2π3+π4=(11π4+π4)−2π3=3π−2π3=7π3\theta_{net} = \frac{11\pi}{4} - \frac{2\pi}{3} + \frac{\pi}{4} = \left(\frac{11\pi}{4} + \frac{\pi}{4}\right) - \frac{2\pi}{3} = 3\pi - \frac{2\pi}{3} = \frac{7\pi}{3} radians.
Grouping terms with the same denominator simplifies the fraction arithmetic.
3
Find the coterminal angle of 7π3\frac{7\pi}{3} in the interval [0,2π)[0, 2\pi).
7π3=2π+π3\frac{7\pi}{3} = 2\pi + \frac{\pi}{3}, which is coterminal to π3\frac{\pi}{3} radians.
Subtracting integer multiples of 2π2\pi (full revolutions) gives the standard position of the terminal ray.
4
Evaluate the coordinates of the point on the unit circle at π3\frac{\pi}{3} radians.
(x,y)=(cos⁡(π3),sin⁡(π3))=(12,32)(x, y) = \left(\cos\left(\frac{\pi}{3}\right), \sin\left(\frac{\pi}{3}\right)\right) = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right).
For any angle θ\theta on the unit circle, the coordinates are given by (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta).

Anahtar Kavram

Unit circle coordinates, angle conversion, and coterminal angles
Tahmini Süre:2m 0s
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Konular

Unit Circle and Angle Measures (Radians and Degrees)

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