Question

the point p(x, y) lies on the terminal side of an angle $\theta = \frac{3\pi}{4}$ in standard position. what are the signs of the values of x and y?
\bigcirc both x and y are positive.
\bigcirc both x and y are negative.
\bigcirc x is positive, and y is negative.
\bigcirc x is negative, and y is positive.

Explanation:

Step1: Determine the quadrant of \(\theta = \frac{3\pi}{4}\)

Angles in standard position: \(0 < \theta < \frac{\pi}{2}\) (Quadrant I), \(\frac{\pi}{2} < \theta < \pi\) (Quadrant II), \(\pi < \theta < \frac{3\pi}{2}\) (Quadrant III), \(\frac{3\pi}{2} < \theta < 2\pi\) (Quadrant IV).
\(\frac{\pi}{2} = \frac{2\pi}{4}\), \(\pi = \frac{4\pi}{4}\). So \(\frac{2\pi}{4} < \frac{3\pi}{4} < \frac{4\pi}{4}\), meaning \(\theta = \frac{3\pi}{4}\) is in Quadrant II.

Step2: Recall sign rules for Quadrant II

In Quadrant II, the \(x\)-coordinate (horizontal) is negative, and the \(y\)-coordinate (vertical) is positive. For a point \(P(x, y)\) on the terminal side, \(x < 0\) and \(y > 0\).

Answer:

D. \(x\) is negative, and \(y\) is positive.