On the unit circle, what relationship do the ratios $\frac{\sin x}{1}$ and $\frac{\cos y}{1}$ share for any real values of $x$ and $y$?

• A. Both $\sin x$ and $\cos y$ represent the y- and x-coordinates, respectively, of points on the unit circle.
• B. Their sum is always $1$.
• C. They are always equal for all values of $x$ and $y$.
• D. Their product is always $0$.

Answer: (A)

Verified step by step guidance

1. Recall that on the unit circle, any point can be represented as \((\cos \theta, \sin \theta)\), where \(\theta\) is the angle formed with the positive x-axis.
2. Understand that \(\sin x\) corresponds to the y-coordinate of a point on the unit circle at angle \(x\), and \(\cos y\) corresponds to the x-coordinate of a point on the unit circle at angle \(y\).
3. Recognize that both \(\sin x\) and \(\cos y\) are values between -1 and 1, representing coordinates on the unit circle.
4. Note that the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) holds for the same angle \(\theta\), but this does not imply that \(\sin x + \cos y = 1\) or that they are always equal for different angles \(x\) and \(y\).
5. Therefore, the key relationship is that \(\sin x\) and \(\cos y\) represent the y- and x-coordinates, respectively, of points on the unit circle, but no fixed sum or product relationship holds for arbitrary \(x\) and \(y\).