Its center at O = (0, 0), the unit circle x2 + y2 = 1 goes through P = (1, 0). The line y = 0.6 intersects the circle at A and B, with A in the first quadrant. The angles POA and POB are said to be in standard position, because their initial ray OP points in the positive x-direction. (Their terminal rays are OA and OB.) Find the sizes of these angles.
How are they related?
Then (Continuation) If we restrict ourselves to a single revolution, there are actually two angles in standard position that could be named POB. The one determined by minor arc PB is said to be positive, because it opens in the counterclockwise direction. Find its degree measure. The one determined by major arc PB is said to be negative, because it opens in the clockwise direction. Find its degree measure.
To find the coordinates of point A, we plug y=0.6 into the equation of the unit circle:
x^2 + (0.6)^2 = 1
x^2 + 0.36 = 1
x^2 = 0.64
x = ±0.8
So point A has coordinates (0.8, 0.6) (since it's in the first quadrant) and point B has coordinates (-0.8, 0.6).
Using the coordinates of point A, we can find the cosine and sine of angle POA:
cos(POA) = 0.8
sin(POA) = 0.6
To find the angle POA, we can use the inverse cosine function:
POA = arccos(0.8) ≈ 36.87°
Similarly, we can find the angle POB:
cos(POB) = -0.8
sin(POB) = 0.6
POB = arccos(-0.8) ≈ 143.13°
The angles POA and POB are related because they are supplementary, meaning their sum is equal to 180°.
For the positive angle POB, the degree measure is 143.13° (as found above) since it refers to the angle determined by minor arc PB and opens in the counterclockwise direction.
For the negative angle POB, we need to find the angle determined by major arc PB and opening in the clockwise direction. To do this, we can simply subtract the positive angle POB from 360°:
Negative angle POB = 360° - 143.13° ≈ 216.87° (in negative direction)
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°.
To find the sizes of angles POA and POB, we can use trigonometry and the properties of the unit circle.
Angle POA:
Since P = (1, 0) lies on the x-axis and O = (0, 0) is the center of the circle, OA is the radius of the circle, which measures 1 unit. Thus, we have a right triangle with O, P, and A as its vertices. We can use the Pythagorean theorem to find the length of side PA:
PA^2 = OA^2 - OP^2
PA^2 = 1^2 - 1^2
PA^2 = 0
PA = 0
Since PA has zero length, triangle OPA degenerates into a straight line. Therefore, angle POA measures 180 degrees or π radians.
Angle POB:
To find the size of angle POB, we need to determine the corresponding point B on the circle where the line y = 0.6 intersects it. To find B, we can substitute y = 0.6 into the equation x^2 + y^2 = 1:
x^2 + (0.6)^2 = 1
x^2 + 0.36 = 1
x^2 = 1 - 0.36
x^2 = 0.64
x = ±√0.64
Since A is in the first quadrant, B will have a positive x-coordinate. Thus, x = √0.64 = 0.8. Hence, point B is (0.8, 0.6).
To find angle POB, we can use the formula for the slope of a line passing through two points:
m = (y2 - y1) / (x2 - x1)
m = (0.6 - 0) / (0.8 - 1)
m = 0.6 / -0.2
m = -3
The slope of the line passing through O and B is -3. To find the angle POB, we can take the arctangent of the slope:
POB = arctan(-3)
Using a calculator, we find that POB ≈ -71.57 degrees or -1.249 radians.
To find the positive angle determined by the minor arc PB, we simply add 360 degrees or 2π radians to the negative angle:
Positive angle = -71.57 degrees + 360 degrees = 288.43 degrees
Positive angle ≈ 288.43 degrees or 5.04 radians.
To find the negative angle determined by the major arc PB, we subtract 360 degrees or 2π radians from the positive angle:
Negative angle = 288.43 degrees - 360 degrees = -71.57 degrees
Negative angle ≈ -71.57 degrees or -1.249 radians.
Therefore, the positive angle determined by the minor arc PB is approximately 288.43 degrees or 5.04 radians, while the negative angle determined by the major arc PB is approximately -71.57 degrees or -1.249 radians.