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Identify the curve r = 1/(2 + sin 0) and find its parameters a.b,€ d.e (b) Write aH equivalent Cartesian equation:...

Question

Identify the curve r = 1/(2 + sin 0) and find its parameters a.b,€ d.e (b) Write aH equivalent Cartesian equation:

Identify the curve r = 1/(2 + sin 0) and find its parameters a.b,€ d.e (b) Write aH equivalent Cartesian equation:



Answers

Identify the curve by finding a Cartesian equation for the curve

$ r^2 \sin 2\theta = 1 $

Okay, so we have this polar question and let's convert this into direct angular coordinates. Okay, okay, so let's expand this stunning expression. Okay, so I got Karzai of Pie is -1. The sine of pi cost zero. Okay, So I got r equals -4 times I of say to So let's multiply our to both sides of the equation. Got r squared equals but it's for our time. Side of data with the R squared is just X squared plus y square, Which people sweat is four times. Why? Okay. Uh huh. So X squared plus Y squared Plus four. Y equals zero. So this is the equation of a circle. Okay, we can rewrite this as X squared plus why? Plus two squared equals four. Okay. Yeah, so this is a rectangular form of the polar equation and the shape is a circle. This curve is a circle.

So we have a polar equation. R equals two Times co sign of data that we want to convert it to a Cartesian equation. So let's recall that X equals R Times Co sign of data. So we divide both sides of this equation. Where are you? See that X over our is equivalent to co sign of data. So by direct substitution, we have our equals two times, replacing co sign of data with X over heart, we cannot multiply those on both sides of our equation. By our toe, obtain R squared equals two times X. Next, plus recall that are squared is equal to X squared plus y square so we can replace R squared with X squared plus y squared, still equaling two X. And though this is a Cartesian equation, we want to now write it in such a way that it it's clear in terms of what type of curve this equation would represent. So let's subtract two X on those sites, and as we do that out, group the X terms together. Next, we want to complete the square on the excess by adding in negative two divided by two quantity squared. We know the words negative one square, which is one. So I'm gonna add one on both sides of the equation to keep the equation balance and to complete the square next would want a factor into perfect squares. Now we can see that this has the general form of the equation. Hello, Circle. Therefore, there's a circle century at one comma zero with the radius one.

For this set of problems, We're going to be discussing how you eliminate a parameter. So eliminating a parameter is uh not only easy, it can be difficult but it is also very effective. So one way that we can do this, for example is by letting X equal some value. We'll have this case of X equaling the sign fx and we'll have y equal I agree the co sign, I have one half um sorry T. Mexico 70. And this. Okay. So the way that we eliminate the parameter is we solve for T. So what we're gonna do is we're going to solve for T. By taking the inverse sine of X. And that's in our our key value and then we can input that right here. So when we do that we end up getting this right here as the function and that's going to be how we eliminate perimeter.

Okay, so You have this equation articles two times I of Ceta. Okay, so let's convert this into a rectangular question. Okay, So first let's add multiple are to the both sides of the equation. So I have r squared equals two are inside of Ceta. Okay, so we know R squared is actually X squared plus y squared. Okay. And what is two times are times sine of Sita's? Our time say of Sita is why it goes to why? So we know this is uh, equation of a circle. Okay?


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