Question

which of the following is true of the values of x and y in the diagram below? (diagram description: a unit circle with points (0,1), (-1,0), (1,0), (0,-1), and a point (x,y) at angle θ = π/4 with radius r=1)

Explanation:

Step1: Recall Unit Circle Definitions

On the unit circle (radius \( r = 1 \) here, though the diagram shows \( r = 2 \), we can use trigonometric relationships. For an angle \( \theta \) in standard position, \( x = r\cos\theta \) and \( y = r\sin\theta \). Here, \( \theta=\frac{\pi}{4} \) and \( r = 2 \).

Step2: Calculate \( x \) and \( y \)

Using \( x = r\cos\theta \) and \( y = r\sin\theta \) with \( r = 2 \) and \( \theta=\frac{\pi}{4} \):

• \( \cos\frac{\pi}{4}=\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2} \)
• \( x = 2\cos\frac{\pi}{4}=2\times\frac{\sqrt{2}}{2}=\sqrt{2} \)
• \( y = 2\sin\frac{\pi}{4}=2\times\frac{\sqrt{2}}{2}=\sqrt{2} \)

So \( x = y \) (since both equal \( \sqrt{2} \)).

Answer:

\( x = y \) (or specific true statement like \( x \) and \( y \) are equal, depending on context)