Convert the polar equation to a rectangular equation. You do not need to solve for y. Make sure to include the = sign.

Question

Convert the polar equation to a rectangular equation. You do not need to solve for . Make sure to include the = sign. 

Polar equation is 

Rectangular equation is

Answer 1

When converting between polar coördinates (r, θ) and rectangular coördinates (x, y), the Pythagorean theorem gives us the conversion

r = √(x² + y²)

Squaring both sides of the above gives us

r² = x² + y²

The definitions of sine and cosine give us

y = sin θ

x = cos θ

We will use these because the tan θ in the problem is merely opposite/adjacent = (sin θ)/(cos θ).

Therefore the polar equation

r² = 10 tan θ

(which is really just r² = 10 (sin θ)/(cos θ))

becomes the rectangular equation

x² + y² = 10y/x

Answer 2

rectangular equation: x² + y² = 10y/x

Answer 3

x^2 +y^2 = r^2

tanT = sinT/cosT = (y/r)/(x/r)= y/r

10tanT = 10y/x

r^2 =10tanT

x^2+y^2 = 10y/x

although it might be better to rewrite it as

x^3+xy^2 = 10y

as the polar equation includes the origin

while the 10y/x excludes it

and the

x^3 +xy^2 =10y includes it

r^2 is always >0

tanT is only >0 in quadrants I and III

the cubic in rectangular coordinates is also only in quadrant I and III