Polar Coordinates

Question

A curve in polar coordinates is given by: r=8+5costheta. Point P is at theta = 21pi/16.

a) Find the coordinate r for P, with r>0 and pi <theta < 3pi/2.

b) Find the cartesian point for P.

x=?, y=?

c)  How may times does the curve pass through the origin when 0< theta < 2pi.

I am having trouble solving this and I do not know where I am making my mistake. If you could go through the steps, that would be great!

Answer 1

I don't blame you. This question is poorly worded at best.

a)

Plug θ = 21π/16 into r = 8 + 5cosθ and get r = 5.222.

The statement of the problem is confusing as it seem to place a constraint

on both r and θ, although both are fixed by the placement of the point P.

Either somebody is trying to confuse you, or make you think about the placement of P.

The constraint π < θ < 3π/2 points out that P is in the third quadrant,

where cosθ < 0.

If, hypothetically, the curve were r = 5cosθ, that would make r < 0, so

you would need to reverse the sign of r and place P in the first quadrant.

But with your given curve none of this is necessary.

b)

Just plug into the usual formulas, getting cartesian coordinates in the third quadrant.

x = rcosθ = -2.9

y = rsinθ = -4.34

c)

The curve passes through the origin for those θ, which make r = 0.

This never happens, because for all θ

cosθ ≥ -1

So

8 + 5cosθ ≥ 3