Convert to an equation in rectangular coordinates. r= 6csc(θ)−sec(θ) ​ (Express numbers in exact form. Use symbolic notation and fractions where needed.)

Question

Convert to an equation in rectangular coordinates. r= 6csc(θ)−sec(θ) ​ (Express numbers in exact form. Use symbolic notation and fractions where needed.)

Answer 1

Let's first convert those secant and cosecant into cosine and sine.

r = ⁶/_sinθ - ¹/_cosθ

Next, we can divide both sides by r.

1 = ⁶/_r•sinθ - ¹/_r•cosθ

y = r•sinθ and x = r•cosθ, so let's substitute those in.

1 = ⁶/_y - ¹/_x

To solve for y, we could first multiply everything by xy to get rid of the fractions. After that, it's just general algebra.

xy = 6x - y

y(x+1) = 6x

y = ^6x/_x+1

Hope that helps. I know it's been a week since you asked.

Answer 2

r= 6cscT - secT

1 = 6cscT/r - secT/r

use identities:

y= rsinT= r/cscT,

(cscT)/r = 1/y

x=rcosT=r/secT,

(secT)/r = 1/x

1= 6/y - 1/x

or

multiply by xy to eliminate fractions

xy=6x-y

or solve for y

(x+1)y =6x

y = 6x/(x+1)

f(x) = 6x(x+1)