Can someone please help me with converting from rectangular equations to polar form?

Question

Basically my pre-calc teacher gave the class this homework packet, and i don't know how to do 5/36 of them. I need help.

There are the problems:

Convert the rectangular equation to polar form.

1. y^2 = 2x

2. y^2 = x^2

3. x^2 = y^3

4. (x^2 + y^2)^2 - 9(x^2 - y^2) = 0

5. y^2 - 8x -16 = 0

Any help would be much appreciated.

Answer 1

All you have to know to go one way is:

y = r sin(theta)
x = r cos(theta)

the other way is

r = sqrt(x² + y²)
theta = arctan (y/x)

so just substitute and simplify:

y² = 2x
(r sin(theta))² = 2(r cos(theta))
r² sin²(theta)) = 2r cos(theta)

divide both sides by r and sin²(theta) to make it look nicer:

r = 2cos(theta)/sin²(theta)

Or even simpler, using trig identities:

r = 2cot(theta)csc(theta)

Enjoy!

Answer 2

you may convert from oblong to polar utilizing the formula: X = R*cos? Y = R*sin? in case you enhance the (y-5)^2 area of the expression you wrote, you get: X^2 + Y^2 + 25 - 10Y = 25 in case you plug the polar variations of X and Y into this expression, you get: R^2 * cos^2(?) + R^2 * sin^2(?) + 25 - 10R*sin? = 25 component out the R^2 words to get: R^2 *( cos^2(?) + sin^2(?) ) + 25 - 10R*sin? = 25 The cos^2 + sin^2 area is an identical as one, so that you've: R^2 - 10R*sin? = 0 it really is an similar as: R = 10sin?