Question

plot the point \\( \left(4, \frac{3 \pi}{4}\
ight) \\) given in polar coordinates, and find other points \\( (r, \theta) \\) of the point for which the following are true
(a) \\( r>0,-2 \pi \leq \theta<0 \\)
(b) \\( r<0,0 \leq \theta<2 \pi \\)
(c) \\( r>0,2 \pi \leq \theta<4 \pi \\)
select the graph that represents the point \\( \left(4, \frac{3 \pi}{4}\
ight) \\)
oa.
ob.
oc.

Explanation:

Step1: Analyze polar coordinates rules

For a polar coordinate \((r,\theta)\), when \(r>0\), the point is in the direction of \(\theta\) from the origin. When \(\theta =-\frac{3\pi}{4}\), we can think of it as rotating clockwise \(\frac{3\pi}{4}\) from the positive \(x -\)axis.

Step2: Check each option

• Option A: The graph in option A does not correspond to the angle \(-\frac{3\pi}{4}\) with \(r = 4>0\).
• Option B: The graph in option B does not correspond to the angle \(-\frac{3\pi}{4}\) with \(r = 4>0\).
• Option C: When \(r = 4>0\) and \(\theta=-\frac{3\pi}{4}\), we start from the positive \(x -\)axis and rotate clockwise \(\frac{3\pi}{4}\) radians. The length from the origin \(r = 4\) matches the requirements of \(r>0,-2\pi\leq\theta<0\).

Answer:

C.