In Exercises 49–58, convert each rectangular equation to a pol...

Question

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.

x² + (y + 3)² = 9

Accepted Answer

Welcome back, everyone. Transform the given a rectangular equation X2 + Y + 82 equals 25 into a polar equation expressing R in terms of theta. And we're given 4 answer choices. A says R2 equals 16 sin theta minus 39. B2 equals -16 are sin theta minus 39 C R 2 equals 16 are sin theta minus 39, and dr 2 equals 16 are sin theta plus 39. So let's go ahead and solve this equation. Let's recall that x is equal to R cosine theta. And Y is equal to R sine theta. And what we're going to do is basically substitute these into the given equation. So we end up with R cosine theta squared. Plus, now Y is equal to R sine theta and we're going to add 8 and then square it. This must be equal to 25. And now let's square the terms. The first one becomes R squared. Cosine squared theta. And for the second one, we can apply the formula. First of all, we square the 1st term and this gives us R squared. Sin square theta. Then we multiply the two terms. And multiply by 2. So this gives us 2 multiplied by 8. Which gives us 16 and then multiplied by our sin theta. And finally, we square 8, right? So we are going to add 64, and this must be equal to 25. Now our goal is to express R in terms of theta. So first of all, we can factor out our squared and this gives us Sign Squared theta plus cosine squared theta. And then we have plus. 16 R. Sign data, and now if we subtract 25 from both sides, we're also going to get + 39. This must be equal to 0. Now, sin squared theta plus cosine squared theta is equal to one, right, based on Pythagorean identity. And now we can isolate R squared. So R squad is going to be equal to, now we're going to move the two terms to the right hand side, specifically we're going to get negative 16. Are sign theta. And then we're going to get 39, right? We are going to flip those signs, and that's it. As we can see, the final answer to this problem corresponds to the answer choice B. Thank you for watching.

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.

x² + (y + 3)² = 9

Key Concepts

Rectangular to Polar Coordinates

Equation of a Circle

Trigonometric Identities

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