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Question 2_ The vale of the integralCOS sin z _2tiwhcre is the circle |z| = 8, oriented counterclockwise; is equal...

Question

Question 2_ The vale of the integralCOS sin z _2tiwhcre is the circle |z| = 8, oriented counterclockwise; is equal

Question 2_ The vale of the integral COS sin z _ 2ti whcre is the circle |z| = 8, oriented counterclockwise; is equal



Answers

Find zw and $\frac{z}{w}$. Leave your answers in polar form. $$ z=2\left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right) $$

Okay, so we're given the first point. Z is for co sign of three pi over eight. Plus I sine of three pi over eight. And the second point is going to be to co sign of nine pi over 16. Plus I sign of nine pi over 16 okay? And we're looking to find the division of Z minus Z over W. So the first thing that we need to do is we need to divide the the Ma July here. And so we've got four divided by two, which is two. And then because we're doing division, we're gonna subtract the arguments. So I have three pi over AIDS minus nine pi over 16. Plus I sine of three pi over eight, minus nine pi over 16. Okay, so in order to do this, we need a common denominator. So we've got three pi over eight, minus nine pi over 16. Common denominators. Sixteens. We've got six pie or 16 minus nine pi over 16. Right. Which is gonna give us negative three pi over 16 which is not in the first rotation. So we want to clean that up just a little bit more, So we're gonna add one rotation. We're gonna add one cycle, which is two pi. Well, again, we're gonna need a common denominator. So I got negative three pi over 16 and I need to have a common denominator of 16. I need 32 pies inward, equal to pi. And so when I put that together, I'm gonna end up with 29 pi over 16 which would be the argument in the first rotation. So let's just clean up our answer here. We've got two times the co sign of 29 pi over 16 plus I sine of 29 pi over 16. And this is the best answer that you should have for

Okay, This question asks us to multiply and divide these two complex numbers. So since there in polar form already weaken, just do Z w which multiplying the magnitudes. Together we get a and then we add the angles. So we get three pi over eight plus nine pi over 16. So a common denominator would be by over 16. So we get nine plus six or 15 pi over 16. So that's our angle for the product. And since this is between zero and two pi, we don't have to do anything to our angle. We can just leave it like that. Then for the quotient, we start by dividing the magnitudes. So four divided by two which is to and then we subtract the angles. So we have co sign, uh, six pi over 16 minus nine pi over 16. So negative three pie over 16 then Plus I sine negative three pi over 16. And we like our angles to be between zero and two pi. So we can just add two pi to this, which is the same thing as 32 Hi, over 16. So we get co sign of negative three plus 32 which is 29 pie over 16 plus I times sign of 29 pie over 16 and that's our answer.

Were given the effect er, field F and an oriented counterclockwise as viewed from above curve see and were asked. He Stokes his theory, um, to evaluate the line Integral oversee of F So f is the vector field y Z I plus two times x times e j plus e to the X y que and C is the Circle X squared plus y squared equals 16 z equals five. So first of all, it's calculate the curl of our vector field f This is X times E to the X y minus two x I minus Why times e to the X y minus Why, j plus to Z minus z okay. And we'll take the surface s to be the disk X squared plus y squared is less than or equal to 16 z equals five. This is the surface bounded by our curve. Now see is oriented counter clockwise from above. Therefore, it follows that but want to orient the surface s upward, then our normal vector n will simply be equal to okay and we'll have that The curl of f started with n simply going to be the K component of the curl which is to Z minus Z, which is simply Z and in particular this is equal to five on the surface s since C is equal to five. Therefore, we have that the line integral oversee of F by Stokes theorem. This is equal to the surface integral Over s of the curl of f which is equal to the double integral over s of curl of F dotted with n which was simply five ds. So this is equal to five times the area of the surface s which were called was just a disk of radius four. This is five times high times four squared, which is the same as 80 pie.

Okay. Wishing given in this pushes excess crap. That's my spot. Yeah. So determine with minus routine Truth car so steep that his presidency sincerely on this so called there existed party. So we have to find out the back door, everybody a spot. Today's decision. She got your phone. Second quarter look solid from York. Chicharito equals minus. Find my tree. And if you simplify Tiu, Tiu and Kenny See this? This is love. And in order to see how it started or didn't you Just negative for me? Well, since she usually alone. So some cactus. Okay, they Oh, no. If you see as you already know, they're excellent article. And what did you do? Levi's Lexus Park Close. More sparring when you hear Borders eight moving. So there's this movie of the year for So are you, Tito. Okay. So that at that point, you to do also my this to my Ritchie when Mr By Ricky. Okay. Next time you're faced with the same via chain. Oh, and course where this to wait three. Do you? Course, my Mr Fiber King. Actually, 66. This will be my guest when they go on disability is also went into this. Maybe you buy this, and if it's in your face, I believe where you miss six for more on this part.


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