Final answer:
The points on the unit circle where the sine value is 1/2 are at angles π/6 and 5π/6, which correspond to the coordinates (sqrt(3)/2, 1/2) and (-sqrt(3)/2, 1/2) respectively.
Step-by-step explanation:
The student is asking for the points on the unit circle where the sine value is 1/2. On the unit circle, the sine of an angle corresponds to the y-coordinate of the point where the terminal side of the angle intersects the circle. Since the unit circle has a radius of 1, for sine to be 1/2, we are looking for angles where the y-coordinate is 1/2.
To find these points, you can use the fact that the sine function oscillates between +1 and -1 every 2 radians on the unit circle. In the first quadrant, the y-coordinate of a point on the unit circle is equal to the sine of the angle formed with the positive x-axis.
So, for the angle θ where sin(θ) = 1/2, you can find the corresponding x-coordinate by using the Pythagorean theorem: x² + (√3/2)² = 1. Solving for x, you get x = ±√3/2. Therefore, the points on the unit circle with a sine value of 1/2 are (1/2, √3/2) and (-1/2, -√3/2).
We can recall that the principal angles where the sine value is 1/2 are π/6 (or 30 degrees) and 5π/6 (or 150 degrees) in standard position. These points correspond to the coordinates (sqrt(3)/2, 1/2) and (-sqrt(3)/2, 1/2) respectively on the unit circle.