To find the sizes of angles POA and POB, we need to first locate points A and B where the line y = 0.6 intersects the unit circle x^2 + y^2 = 1.
1. First, substitute y = 0.6 into the equation of the unit circle:
x^2 + (0.6)^2 = 1
x^2 + 0.36 = 1
x^2 = 1 - 0.36
x^2 = 0.64
x = ±√0.64
x = ±0.8
Hence, the line y = 0.6 intersects the unit circle at two points: A = (0.8, 0.6) and B = (-0.8, 0.6).
2. Next, we can find the angles POA and POB by using trigonometry. The angle at the center of the circle, O, is the reference angle for both POA and POB.
To find angle POA:
We can use the arctan function to find the angle POA in standard position. The arctan function gives the ratio of the opposite side to the adjacent side of a right triangle.
Let's consider the triangle OPA. We can calculate the opposite side length as the difference in y-coordinates (0.6 - 0 = 0.6) and the adjacent side length as the difference in x-coordinates (0.8 - 0 = 0.8).
Using the arctan function, we can find the angle POA:
POA = arctan(opposite/adjacent) = arctan(0.6/0.8)
To find angle POB:
Similarly, we can use the arctan function to find the angle POB in standard position. Consider the triangle OPB. The opposite side length is still 0.6, but the adjacent side length is now the absolute value of the difference in x-coordinates (|-0.8 - 0| = 0.8).
Using the arctan function, we can find the angle POB:
POB = arctan(opposite/adjacent) = arctan(0.6/0.8)
Now, based on the information given, we can proceed with the continuation of the problem.
3. To find the degree measure of the positive angle determined by minor arc PB, we need to determine the angle that PB subtends at the center of the circle, O.
Since PB is a chord, and the unit circle has a radius of 1, PB forms an isosceles triangle with the center, O. We can consider the triangle OBP, where the sides OB and OP have equal lengths of 1.
Since the triangle is isosceles, the angle at the center, O, is twice the angle at P or B. Let's denote the angle at P or B as x.
2x + x = 180° (since the angles of a triangle add up to 180°)
3x = 180°
x = 180° / 3
x = 60°
Therefore, the positive angle determined by the minor arc PB is twice the angle at P or B, which gives us 2 * 60° = 120°.
4. To find the degree measure of the negative angle determined by major arc PB, we take the supplement of the positive angle. The supplement of an angle is the amount needed to make it add up to 180°.
Hence, the negative angle determined by the major arc PB is 180° - 120° = 60°.