5. given f(θ)=−1−sinθ. as θ increases on interval (π,3π/2), which of the following statements is true about…

Explanation:

Step1: Analyze the sign of (y = r\sin\theta)

We know (r=- 1-\sin\theta) and (y = r\sin\theta=(-1 - \sin\theta)\sin\theta=-\sin\theta-\sin^{2}\theta). For (\theta\in(\pi,\frac{3\pi}{2})), (\sin\theta\in(- 1,0)). Let (t = \sin\theta), then (y=-t - t^{2}=-(t^{2}+t)=-(t+\frac{1}{2})^{2}+\frac{1}{4}). Since (t\in(-1,0)), (y<0), so the points are below the horizontal - axis.

Step2: Analyze the change of (|r|)

We have (r=-1-\sin\theta). Take the derivative of (r) with respect to (\theta), (r'=-\cos\theta). For (\theta\in(\pi,\frac{3\pi}{2})), (\cos\theta\in(0,1)), so (r'=-\cos\theta<0). Also, (r=-1 - \sin\theta), when (\theta) increases from (\pi) to (\frac{3\pi}{2}), (\sin\theta) increases from (0) to (-1). (r=-1-\sin\theta) increases from (- 1) to (0) (in magnitude, (|r|) is getting closer to the origin).

Answer:

C. Points are below the horizontal - axis and getting closer to the origin.