Convert each equation to its polar form.x2+y2=2yx^2+y^2=2yx2+y2=2y

r = 2 sin ⁡ θ r=2\sin\theta r = 2 sin θ

Convert each equation to its polar form. x 2 + y 2 = 2 y x^2+y^2=2y x 2 + y 2 = 2 y

Here we're asked to convert the equation into its polar form and the equation that we're given is x squared plus y squared is equal to 2 y. So let's go ahead and dive in here by replacing x y and x squared plus y squared using these formulas. Now remember that if you have an x squared plus a y squared squared, you can simply replace this with r squared instead of using r cosine theta and r sine theta. So on that left side, all I have is r squared, and then this will be equal to 2 times r sine theta replacing that y value. Now from here, I wanna solve for r. So let's look at what's happening here. On both sides, I have an r value. On that left side, I have r squared, and on that right side, just an r. So that r on the right will cancel out with 1 of the r squareds on the left. Now that just leaves me with r on that left side and this is equal to 2 times the sine of theta and this gives me my final answer. This is my equation in its polar form. Let me know if you have any questions. Thanks for watching.

Convert each equation to its polar form. x 2 + y 2 = 2 y x^2+y^2=2y x 2 + y 2 = 2 y

r = 2 sin ⁡ θ r=2\sin\theta r = 2 sin θ

r = 2 r=\sqrt{2} r = 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

r = 4 sin ⁡ θ r=4\sin\theta r = 4 sin θ

r = 2 cos ⁡ θ r=2\cos\theta r = 2 cos θ

9. Polar Equations

Convert Equations Between Polar and Rectangular Forms

Trigonometry

9. Polar Equations

Convert Equations Between Polar and Rectangular Forms

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Multiple Choice

Convert each equation to its polar form.
$x^2+y^2=2y$

$r=\sqrt{2}$

$r=2\sin\theta$

$r=4\sin\theta$

$r=2\cos\theta$

Verified step by step guidance

Start by recalling the relationship between Cartesian coordinates (x, y) and polar coordinates (r, θ). The formulas are: x = r\cos\theta and y = r\sin\theta.

Substitute x = r\cos\theta and y = r\sin\theta into the given equation x^2 + y^2 = 2y.

The equation becomes (r\cos\theta)^2 + (r\sin\theta)^2 = 2(r\sin\theta).

Simplify the left side using the identity \cos^2\theta + \sin^2\theta = 1, which gives r^2 = 2r\sin\theta.

Divide both sides by r (assuming r ≠ 0) to isolate r, resulting in r = 2\sin\theta.

3:37m

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