Question

Let x = 33/65 be the x-coordinate of the point P(x, y) where the terminal side of angle θ (in standard position) meets the unit circle. If P is in Quadrant IV, what is sin(θ)?

a) sin(θ) = √(1 - x²)
b) sin(θ) = -√(1 - x²)
c) sin(θ) = -x
d) sin(θ) = x

Answer 1

The value of sin(θ) for a point P(x, y) in Quadrant IV on the unit circle with x-coordinate 33/65 is -√(1 - x²), as sine is negative in the fourth quadrant.

Step-by-step explanation:

To find the value of sin(θ) where P(x, y) is the point on the unit circle corresponding to the angle θ in standard position, with P being in Quadrant IV and x given as 33/65, we use the Pythagorean identity sin²(θ) + cos²(θ) = 1. Since P is in Quadrant IV, sin(θ) will be negative because sine is negative in that quadrant. To find the exact value, we calculate:

• sin²(θ) = 1 - cos²(θ) = 1 - x² = 1 - (33/65)²
• sin(θ) = -√(1 - x²) because sin(θ) is negative in Quadrant IV

Thus, the correct option is b) sin(θ) = -√(1 - x²).