pre calc word problem

Question

A curve in polar coordinates is given by r=7+2cos∅. point P is at ∅=16π/14

a.) find polar coordinate r for P with r>0 and π<∅<3π/2.

r=?

b.) find Cartesian coordinates for point P

x=?

y=?

c.) How many times does the curve pass through the origin when 0< ∅<2π?

Answer 1

(Note: I'm using φ instead of ∅ as you do. Nothing about the meaning really changes. I just prefer "my phi" φ because ∅ generally denotes the empty set. Also, θ is generally preferable to φ because there are later contexts where φ has a specific meaning that isn't the same as θ).

(a). The polar coordinate r for the given φ, namely φ=16π/14=8π/7, is just given by plugging that φ value into your given r equation: r=7+2cosφ. So, r=7+2cos(8π/7). Since 8π/7 isn't a nice unit circle angle, unless the question asks you for a decimal approximation, you can just leave your r like that.

(b). The conversion between polar and rectangular/normal/xy-coordinates is made with the following "dictionary":

1. x=rcosφ
2. y=rsinφ
3. r=sqrt{x^2+y^2}
4. φ is either tan^{-1}(y/x) if x>0, -π/2 if x=0 and y<0, π/2 if x=0 and y>0, or π+tan^{-1}(y/x) if x<0.

(c). A curve goes through the origin *exactly* when r=0. But look at how r is defined for this curve: for any value of φ, -1≤cosφ≤1, so -2≤+2cosφ≤2, meaning 5≤r=7+2cosφ≤9. In particular, r is NEVER 0 on this curve, so it NEVER passes through the origin. (If that 7 were a 0, 1, or 2 instead, though, you'd pass through the origin, probably more than once: you figure out "when"/how by solving the equation r=0 for the appropriate values of φ)

Answer 2

a. r = 7+2cos(8Π/7) = 5.198

b.

(x,y)=(7cos(2cos(π7)),−7sin(2cos(π7)))

c. none