Imagine the unit circle with a circle. The circle has center Q(0,0) and P(1,0) lies on it. Another point S(1//2,y) lies on the unit circle. Which of the following could be a possible measure of angle PQS? A. -709 pi//6 B. -719 pi/6 C. -319 pi//3 D. -309 pi//3

Question

Imagine the unit circle with a circle. The circle has center  Q(0,0) and  P(1,0) lies on it. Another point  S(1//2,y) lies on the unit circle. Which of the following could be a possible measure of angle PQS? A.  -709 pi//6 B. -719 pi/6 C.  -319 pi//3 D.  -309 pi//3

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Understand the problem: Understand the problem. We need to find the measure of angle PQS on the unit circle where point Q is the center (0,0), point P is on the circle at (1,0), and point S is on the circle at (1/2, y). We are given four options for the measure of angle PQS in terms of pi.Identify circle properties: Identify the properties of the unit circle. On the unit circle, the angle is measured in radians from the positive x-axis (point P) counterclockwise. The full circle is 2pi radians, so any angle can be expressed as a multiple of pi.Analyze given options: Analyze the given options. We need to determine which of the given options could represent the angle PQS. Since the angle is measured from the positive x-axis, negative angles indicate a clockwise direction. We are looking for an angle that would place point S at (1/2, y) on the unit circle.Convert to positive angles: Convert the options to a positive angle measure. Since angles on the unit circle can be expressed as coterminal angles (angles that share the same terminal side), we can add multiples of 2pi to the negative angles to find a positive coterminal angle.Calculate coterminal angles: Calculate the positive coterminal angles for each option. A. -709 pi/6 + 2pi * k, where k is an integer. B. -719 pi/6 + 2pi * k, where k is an integer. C. -319 pi/3 + 2pi * k, where k is an integer. D. -309 pi/3 + 2pi * k, where k is an integer. We need to find the smallest positive k for each option that makes the angle positive.Find smallest positive k: Find the smallest positive k for each option. A. -709 pi/6 + 12pi/6 * k B. -719 pi/6 + 12pi/6 * k C. -319 pi/3 + 6pi/3 * k D. -309 pi/3 + 6pi/3 * k We need to find the smallest k such that the angle is positive and less than 2pi (since the unit circle is 2pi radians).Calculate smallest positive angle: Calculate the smallest positive k for each option. A. For -709 pi/6, k must be at least 709/12 to make the angle positive. B. For -719 pi/6, k must be at least 719/12 to make the angle positive. C. For -319 pi/3, k must be at least 319/6 to make the angle positive. D. For -309 pi/3, k must be at least 309/6 to make the angle positive.Match angles with point S: Determine the smallest positive angle for each option. A. -709 pi/6 + 709/12 * 12pi/6 = 0 (coterminal with 0) B. -719 pi/6 + 720/12 * 12pi/6 = pi/6 (coterminal with pi/6) C. -319 pi/3 + 320/6 * 6pi/3 = pi/3 (coterminal with pi/3) D. -309 pi/3 + 310/6 * 6pi/3 = pi/3 (coterminal with pi/3)Choose correct option: Match the coterminal angles with the position of point S. Since point S has coordinates (1/2, y), it lies in the first or second quadrant. The angle corresponding to this position must be between 0 and pi. Options C and D give us an angle of pi/3, which is in the first quadrant and could correspond to the position of point S.Choose the correct option. Options C and D are both coterminal with pi/3, which is a possible measure for angle PQS with point S at (1/2, y) on the unit circle. Therefore, both C and D could be correct.

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