polar and cartesian coordinates?

Question

A curve in polar coordinates is given by: r = 8 + 5cos(theta).

Point P is at theta = (21*pi)/16

Find polar coordinate r for P, with r >0 and pi r=
Find cartesian coordinates for point P.
x= ,y=
How may times does the curve pass through the origin when 0

Answer 1

r = 8 + 5 cos(21*pi/16)
x = r cos(theta), y = r sin(theta)

The curve passes the origin when r = 0
or 0 = 8+5 cos(theta) or cos(theta) = -8/5
Since -8/5 is less than -1 then there is no solution for theta.
That means the curve doesn't pass through origin.

Answer 2

If theta = 21pi/16 cos theta = -0.5556 and theta = 4.1233 rad.
r= 8+5(-0.5556)=5.2220
x=r*cos theta =-2.9013

y=r sin theta =-4.3418

For passing through the origin r=0 8 +5 cos theta =0
cos theta = -8/5 possible as cos theta >=-1

Answer 3

theta=pi+5/16pi=
r=8+5(cos(pi+5/16pi))=8-5cos(5/16pi)
x=rcos (pi+5/16pi)=-rcos(5/16pi)
y=rsin (pi+5/16) =-rsin(5/16pi)