$x = \sin \theta, y = \cos \theta, \pi \leq \theta \leq 2\pi$ to rectangular

Question

Given $x = \sin \theta, y = \cos \theta, \pi \leq \theta \leq 2\pi$, a) Eliminate the parameter to find the rectangular equation of the curve; b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.

Is my work below correct?

a) Based on Pythagorean trig identity, $\sin^2{\theta} + \cos^2{\theta} = 1$, $x^2+y^2=1$.

b) Attached is my picture. solution for B

Answer 1

The curve is $x^2 + y^2 = 1, x \leq 0$. Without the restriction you get the entire circle, but you just want a semicircle. Alternatively, $x = -\sqrt{1 - y^2}$.