polar to rectangular and rectangular to polar?

Question

change to polar form

1) 2x-5y^2=7

2) x^4+ xy - 2y^4 = 3y

change to rectangular form

3) r= 4cos(2theta)

4) r= 3 - 3sin(theta)

These 4 problems are giving me some trouble. Can someone help me out on these and explain a little?

Thanks,
Mark

Answer 1

Change to polar form.

Remember the identities.
x = rcosθ
y = rsinθ
x² + y² = r²

1) 2x - 5y² = 7
This is the equation of a parabola.

2x - 5y² = 7
2x = 5y² + 7
5y² - 2x + 7

5r²sin²θ - 2rcosθ + 7 = 0
(5sin²θ)r² - (2cosθ)r + 7 = 0
_______________

Change to rectangular form.

3) r = 4cos(2θ)
This is a four loop flower.

r = 4cos(2θ)
r² = 4r(cos²θ - sin²θ)
r = 4(x²/r² - y²/r²)
r³ = 4x² - 4y²
(x² + y²)^(3/2) = 4x² - 4y²
_____________

4) r = 3 - 3sinθ
This is a cardioid.

r = 3 - 3sinθ
r² = 3r - 3rsinθ

x² + y² = 3√(x² + y²) - 3y
x² + y² + 3y = 3√(x² + y²)

Square both sides.

(x² + y² + 3y)² = 9(x² + y²)

Answer 2

I'll do two of them. You do the other two.
Just keep these 3 rules in mind:
x = r cos(θ)
y = r sin(θ)
x^2 + y^2 = r^2

1.
2x - 5y^2 = 7
2(rcos(θ)) - 5(rsin(θ))^2 = 7
2rcos(θ) - 5r^2 sin^2(θ) = 7

4.
r= 3 - 3sin(θ)

y = rsin(θ)
sin(θ) = y/r

r = 3 - 3(y/r)
r = 3 - 3y/r
r^2 = 3r - 3y
r^2 - 3r + 3y = 0
x^2 + y^2 - 3sqrt(x^2 + y^2) + 3y = 0

Answer 3

i pass to pass forward and take a shot in the lifeless of evening here and wager that your software famous the attitude theta in polar with: theta = arctan(y/x) if that's the case, meaning your software will in basic terms provide values of theta between -pi/2 and pi/2. In different words, your software assumes that x is helpful. while x is detrimental, your software will provide a value of theta it incredibly is off by pi. to repair this, whenever you enter a detrimental value of x, determine you upload pi to the attitude your calculator components, this gives you with the staggering attitude. (in the journey that your software outputs theta in tiers, upload a hundred and eighty instead.)

Answer 4

When do you learn these stuff?!?