In Exercises 59–74, convert each polar equation to a rectangul...

Question

In Exercises 59–74, convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation.

r = 6 cos θ + 4 sin θ

Accepted Answer

Transform the given polar equation into a rectangular equation. Use the resulting equation to draw the graph but we have R equals a cosine theta plus 10 sine theta. And we have a graph here to plot our line up. No, we can first use some substitutions. We know a few substitutions. You know cosign data is X divided by R sign. The is Y divided by R and R squared is equals to X squared plus Y squared. These are just known substitutions probably from our book, but we can use these in our formula here. Let's first substitute our cosign and sign. You get R equals eight X divided by R plus 10 Y divided by R. And we can multiply R on both sides to get rid of the denominators because R squared equals eight X plus 10 Y. Now we can move everything on one side to hit R squared minus eight X minus 10 Y equals zero. Now, we can use our substitution R squared equals X squared plus Y squared. We get X squared plus Y squared minus eight X minus 10 Y equal zero. Now we're noticing this is starting to look like the equation of a circle. We can complete the square to find our full equation of the circle. Let's group together our exits and group together Kara wise now to complete the square, you will take the number of multiplying Rx or ry divided by two and square it and add it to both sides. Other words, we will first take the negative eight on our X divided by two squared that gives us 16. If we erect this, we would get X squared minus eight X and then plus 16, we'll do the same for the Y Y squared minus 10 Y. And for what, we divide our negative 10 by two square. It that gives us 25. We will add 20 facts be a zero plus 16 plus 25 on the right side. Finally, we can turn our X squared minus eight X plus 16 into a single term squared. We can change this into X minus four squared. We'll do the same for Y getting Y minus five squared. All that equals 41. We now have the equation of a circle. The center of a circle will occur at the 0.45. A radius of the circle is the square root of 41 which we can now plot the circle on our graph draw point at 45. And we can just create a circle around, we should then get a circle around that point that looks similar to this with our equation being X minus four squared plus Y minus five squared equals 41. OK? I hope to help you solve the problem. Thank you for watching. Goodbye.

In Exercises 59–74, convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation.
r = 6 cos θ + 4 sin θ

Key Concepts

Polar to Rectangular Coordinate Conversion

Trigonometric Identities and Relationships

Graphing in the Rectangular Coordinate System

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