This Question:31 0Find the slope of the polar curve at the indicated point: 4 cos 0 - 6 sin 0, 0=7OA7 0 B. 3ontentsSuccessJia LibraryOptionsionsToolsCllck solect yo...

$29-48$ Sketch the curve with the given polar equation.
$$r=3 \cos 6 \theta$$

This problem Assets to find slope of the tangent lines for a given polar equation with two different points. So the first step is going to be to take the derivative of this polar questioned er of death data, which is equal to negative three sign data. Then, in order to find some of the tension lines, we can use the equation that d y over D X is equal to d r o ver di fada. Signed data plus our co sign Beta divided by dear of D data Hussein data. My eyes aren't signed data. Then if we plug in d r over di fada, this becomes negative. Three signs square data plus our coastline data divided by negative three Sign theatre Co sign data minus our science data. So the first point we have to solve that is that three playing in these values we get that D wire of DX is equal to negative three sine squared Hi plus three co sign pot divided by negative three sign pie Cosan pie minus 37 pi and selling us out we get find that the denominator is equal to zero, meaning that this is undefined. So the slope at three. Pie is under find. The next point we have to solve a is at 90 taking Do I over the X by playing in 90 gives us negative three sine squared zero plus nine co sign zero divided by a negative three Sign zero co sign zero minus nine. Sign zero. Again, the denominator is equal to zero, meaning that the slope at with the Tanja line at 90 is also undefined.

This problem asks us to solve for the Tanja line of the Given Polar equation at these two different points. So the first step we could take is taking the derivative of the polar equation Diar of Defeat Up which the drug before is just zero. And then the dirt of the science data is CO santa Next noticed all for the tangent line became the equation D y over dx is equal to drv Photo times Scientific data plus r Times Co sign data divided by D r o ver di stato cosign feta minus Farrah Science data then playing in Deir of defeat equals co signed later This will come cosign fate attempts Scient, Ada plus our coastline data divided by cosign square data minus our time Signed data. First, we're gonna solve this at that point for zero and for zero we get the wire. DX is equal to co sign zero sign zero Sorry. Zero plus four co signed zero, but by coastline squared zero minus four. Sign zero. And solving this out gives us for meeting potential in the so potential eyes equal to four at 40 Next. Run into this for the second point, which is at three three pi over two. That gives us that d y over DX this equal to co sign three pi over two signed through power to plus three co sign 33 pi over two, divided by cosign squared three pi over two minus three times. Sign three pi over two and solving this out will give us zero is our final answer and those are our two results.

For this problem. We are asked to find the slope of the tangent line to this curve. It's are equal to six plus three co sign data and were given the points three comma pie and nine comma zero. So we find the slope of the tangent line by using this formula D Y over. The X equals to our prime signed data plus our coast signed data over our prime co sign data plus our sign data outside. This should be minus minus farce and data. So we need to find our prime by deriving are so our prime should be negative three signed data and I am probably that into the formula. Fully she get negative three sign square data plus six co sign Beta plus three co sign square data over negative three. Sign data close. I'm data minus six. Sign data minus three. Sign data co find data and we can simplify. Thus to get um, negative three signs square data plus six coastline data plus three co signs for data over negative six side data close and data minus six. Sign data. And I'm just going to simplify this by dividing all the terms by three to get negative signs. Where Data plus two little sign Data plus co sign. Great data over negative to side data co science data minus two. Sign beta. And now we need Thio. Plug in the values for theta, so I'll do pie first. So, in data equal supply, we should get negative too. Plus one over zero, which is negative. 1/0. So that means that, um, the slip does not exist. That means that it's a vertical line. So now we check for, um when data equals zero and data equals zero, we should get two plus one. Well, zero is three over zero. So again, the slope does not exist. So that means it's a vertical line. And now we check up there. Any point that intercept the origin and we do that by setting are equal to zero. So zero equals 263 co sign data. Some tracks six to both sides. We get negative. Six equals three co sign data. To find both sides by 3 92 equals to co sign beta. And there's not a data where this equation is true. So that means that there are no points that and to accept the origin

A function is given is three sign were zero So a function in terms off Really any sign and the differentiation of this functioning himself? Pity course. No. The X coordinate of this function will be as excessive. Two X course there. Why is it that way? So so off This girl will be devoured by eggs, Okay? And we can also do this. So taking a standard for Mullah off this function as f dish. Keep playing em minus plus f a costea. Do you? Course minus a sign in so in place putting on we get course you don't sign Pity saying Of course. Divided right. Cause in because Peter minus saying Peter into something. Putting the weight off zero course. You know, people like me science, you know, for any same deal keep you get you into cause zero minus you mean signs zero into So the value will come. Any signs? Yes. You know, like, do you think many when you played my nose? So the comes zero divided by me. 00