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 2 over 1 minuteus the sine of theta. Now we want to go ahead and get started with our steps here. And step 1 is to get r cosine theta, r sine theta, or r squared. Now since I'm dealing with sides here, I'm going to go ahead and cancel that out on the right side. And I'm going to go ahead and 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 2. 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 2 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 is 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 2. Now 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 2+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 2 plus y squared. And if I go ahead and multiply this out, I end up with 4 plus 4 y plus y squared. Now 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 4+4y. 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 So here I'm going to go ahead and actually solve for y as we would in a traditional par parabola equation. So in doing that, I'm going to subtract 4 from both sides giving me x squared minus 4 is equal to 4 y. Then if I divide both sides by 4, if I simplify that on the left, I end up with 1 fourth x squared minus 1 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 1 fourth x squared minus 1, 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 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

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 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: Start with the polar equation $ r = \frac{2}{1 - \sin\theta} $. Recall that the conversion from polar to rectangular coordinates involves using the relationships $ x = r\cos\theta $ and $ y = r\sin\theta $.

Step 2: Express $ \sin\theta $ in terms of $ y $ and $ r $ using the identity $ \sin\theta = \frac{y}{r} $. Substitute this into the equation to get $ r = \frac{2}{1 - \frac{y}{r}} $.

Step 3: Simplify the equation by multiplying both sides by $ r $ to eliminate the fraction: $ r^2 = 2r - 2y $.

Step 4: Substitute $ r^2 = x^2 + y^2 $ into the equation to convert it to rectangular form: $ x^2 + y^2 = 2r - 2y $.

Step 5: Rearrange the equation to isolate $ y $ and express it in terms of $ x $: $ y = \frac{1}{4}x^2 - 1 $.

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