Convert each equation to its rectangular form.r=21−sinθr=\frac{2}{1-\sin\theta}r=1−sinθ2

y = 1 4 x 2 − 1 y=\frac14x^2-1 y = 4 1 x 2 − 1

Convert each equation to its rectangular form. r = 2 1 − sin θ r=\frac{2}{1-\sin\theta} r = 1 − s i n θ 2

In this problem we're asked to convert the equation into its rectangular form and the equation that we're given is r equals two over one minus the sine of theta. Now we want to go ahead and get started with our steps here and step one is to get r cosine theta, r sine theta, or r squared. Now since I'm dealing with a fraction here, I want to go ahead and eliminate that fraction by multiplying both sides by my denominator, that one minus sine of theta. Now doing that on both sides here, I'm going to go ahead and cancel that out on the right side and I'm going to distribute that r into that denominator on the left. So that leaves me with r minus r sine theta having distributed that r. Then this is just equal to what I have left on that right side which is two. Now from here I have an r sine theta but I also have an r. Now having an r might not seem ideal, but because I have this r sine theta, I don't really want to do anything else to this equation. So I'm going to go ahead and move on to step two and go ahead and replace everything. Now since I don't have an r squared, I just have an r, how are we going to replace that r? Well, remember that we have x squared plus y squared is equal to r squared. So if I were to square root both sides, I can figure out what r should be replaced with, the square root of x squared plus y squared. So doing that substitution here, I have square root of x squared plus y squared and that r sine theta gets replaced with a y. So subtracting that y, all of that is equal to two. Now since we've replaced everything, we want to go ahead and rewrite our equation in its standard form. Now remember that if we have a square root, we should go ahead and square both sides. But remember that we only want to square both sides when that square root is by itself. So I'm going to go ahead and add y to both sides to get that square root by itself to easily cancel. So on that right side, I now have two plus y. Now I can go ahead and proceed with squaring both sides to eliminate that radical, leaving me with x squared plus y squared is equal to two plus y squared. And if I go ahead and multiply this out, I end up with four plus four y plus y squared. Now let's go ahead and do some simplifying because I have this y squared on both sides that will allow me to cancel it out. Then I'm just left with x squared is equal to four plus four y. Now this does not look like a super familiar equation but because I have this x squared, I want to be thinking about parabolas. So here I'm going to go ahead and actually solve for y as we would in a traditional parabola equation. So in doing that, I'm going to subtract four from both sides giving me x squared minus four is equal to four y. Then if I divide both sides by four, if I simplify that on the left, I end up with one fourth x squared minus one is equal to y. Now we can go ahead and flip this equation around because traditionally these equations are written as y equals something. So here I have a y equals one fourth x squared minus one, just having swapped either side of my equation to end up with now the standard form of a parabola. And now this is my equation in its rectangular form. Let me know if you have any questions here and I'll see you in the next one.

Convert each equation to its rectangular form. r = 2 1 − sin ⁡ θ r=\frac{2}{1-\sin\theta} r = 1 − s i n θ 2

y = 1 4 x 2 − 1 y=\frac14x^2-1 y = 4 1 x 2 − 1

y 2 = 4 − 4 x y^2=4-4x y 2 = 4 − 4 x

x 2 + y 2 = 2 y x^2+y^2=2y x 2 + y 2 = 2 y

x 2 − 1 = y x^2-1=y x 2 − 1 = y

16. Parametric Equations & Polar Coordinates

Polar Coordinates

Calculus

16. Parametric Equations & Polar Coordinates

Polar Coordinates

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Multiple Choice

Convert each equation to its rectangular form.
$r=\frac{2}{1-\sin\theta}$

$y^2=4-4x$

$x^2+y^2=2y$

$y=\frac14x^2-1$

$x^2-1=y$

Verified step by step guidance

Step 1: Understand the polar equation r = 2 / (1 - sin(θ). To convert this to rectangular form, recall the relationships between polar and rectangular coordinates: x = r * cos(θ), y = r * sin(θ), and r^2 = x^2 + y^2.

Step 2: Multiply both sides of the polar equation by (1 - sin(θ)) to eliminate the denominator, resulting in r * (1 - sin(θ)) = 2.

Step 3: Substitute r = √(x^2 + y^2) and sin(θ) = y / r into the equation. This gives √(x^2 + y^2) * (1 - y / √(x^2 + y^2)) = 2.

Step 4: Simplify the equation by distributing √(x^2 + y^2) and combining terms. This will yield an equation involving x and y.

Step 5: Rearrange the equation into a standard rectangular form, such as y = f(x) or x = f(y), depending on the problem's requirements.

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