Final answer:
To find the exact value of cosθ with sinθ = x, use the identity cosθ = √(1 - x²), taking into account the sign of cosine in the specified quadrant to determine if cosθ should be positive or negative.
Step-by-step explanation:
To find the exact value of cosθ, we will use the Pythagorean Theorem which in trigonometric terms states that sin²θ + cos²θ = 1. Given that sinθ = x, we can solve for cosθ using the identity: cosθ = √(1 - sin²θ). Therefore, cosθ = √(1 - x²).
However, since θ is in a specific quadrant, we need to determine the sign of cosθ. In quadrants I and IV, cosine is positive, while in quadrants II and III it is negative. This determination is essential because the square root of a square can be both positive and negative.
If the student meant 'quadrant II' when they wrote 'quadrant y', then since we are in the second quadrant where cosine is negative, the answer would be cosθ = -√(1 - x²). If by 'quadrant y' they meant a different quadrant, you would adjust the sign accordingly.