why is theta in a restricted interval? (Polar coordinates)

Question

What is the polar equation of the circle of radius 1 whose centre lies at the cartesian point (1,0)?

So I got the correct answer of r=2cos(θ) But then is says theta is in the interval (-π/2)≤θ≤(π/2)

Why is this?

Thank you.

Answer 1

The circle is $(x-1)^2+(y-0)^2=1$ so $x^2-2x+1+y^2=1$. Now use $x=r\cos(t),~y=r\sin(t)$ to find that circle in the polar form: $$r^2\sin^2(t)+r^2\cos^2(t)-2r\cos(t)+1=1$$ Or $$r^2-2r\cos(t)=0$$ This means that if $r\neq 0$ then $r=2\cos(t)$. For making sure of what's noted about $t$, just draw that circle. You certainly see that for what values of $t$, the whole plot is being done. It should be restricted.: