Question

Let θ be the measure of an angle, in radians, in standard position with π<θ<3π​/2. Find the exact y-coordinate of the intersection of the terminal side of θ with the unit circle, given cosθ+sin²θ=8159​. State the answer as a single fraction. The exact y-coordinate is___

Answer 1

To find the exact y-coordinate of the intersection of the terminal side of θ with the unit circle, we need to solve the given equation cosθ + sin²θ = 8/159. Since the angle θ is between π and 3π/2, it lies in the third quadrant where the x-coordinate is negative. We can use the sin function to find the y-coordinate.

Step-by-step explanation:

To find the exact y-coordinate of the intersection of the terminal side of θ with the unit circle, we need to solve the given equation cosθ + sin²θ = 8/159. Since the angle θ is between π and 3π/2, it lies in the third quadrant where the x-coordinate is negative. We can use the sin function to find the y-coordinate.

Let's solve the equation:

cosθ + sin²θ = 8/159

(1 - sin²θ) + sin²θ = 8/159

1 = 8/159

sin²θ = 1 - 8/159 = 151/159

sinθ = ±√(151/159)

Since the angle is in the third quadrant, sinθ is negative.

sinθ = -√(151/159)

The exact y-coordinate is -√(151/159).