anonymous 5 years ago express in polar form r cisθ for: a) -1-i b) -3+4i (to 2 decimal places of a degree) c) use your answers to parts a) and b) to simplify (-3+4i)/(-1-i)^2 in polar form.

1. anonymous

first you need $r=\sqrt{a^2+b^2}$ so for part a you get $r=\sqrt{2}$ and for part b you get $r=5$

2. anonymous

then you need $\theta$ which you get by solving $tan(\frac{b}{a})=\theta$

3. anonymous

are you using degree or radians for this? if radians the first one is $\frac{5\pi}{4}$ if degrees is it 225

4. anonymous

if you plot -1-i you see you are in quadrant III which is why you get 225

5. anonymous

for part b you get $tan^{-1}(-\frac{4}{3})$ from a calculator then you have to add 180 to your answer. let me see if i have one

6. anonymous

i get 126.78 to two places but you should check it

7. anonymous

so answer to first one is $\sqrt{2}(cos(45)+isin(45))$ and second is $5(cos(126.78)+isin(126.78))$

8. anonymous

then to divide, you divide the r's and subtract the angles.

9. anonymous

what does it say next to (45)? its all blurry. thanks for your help so far!

10. anonymous

ok first of all i made a mistake. it is not 45 it should be 225

11. anonymous

i don't know why i wrote 45 because earlier i said the angle is 225 degrees

12. anonymous

it says cos(225)+isin(225)

13. anonymous

i guess the latex is hard to read but may be clearer if you refresh the browser, that often works for me

14. anonymous

oh thanks for the tip!

15. anonymous

your next job is to square $(-1-i)$ which is actually very easy not in polar form, but i guess they want it in polar form so we do it

16. anonymous

you do it by squaring r and doubling the angle so you get $(-1-i)^2=\sqrt{2}^2(cos(225\times 2)+isin(225\times2))$ $=2cos(450)+isin(450))$ $=2(cos(90)+isin(90))$

17. anonymous

the last equality because sine and cosine are periodic with period 360 so i just subtracted 360 from 450 to get 90

18. anonymous

btw this problem is really stupid because $(-1-i)^2$ is just 2i and dividing by 2i is easy. very annoying to put in polar form. but in any case we see that since cos(90)=0 and sin(90)=1 we get $(-1-i)^2=2i$ using polar form

19. anonymous

your last job is to divide. as i said dividing by 2i is easy but they want you to use polar form so you divide 5 by 2 and subtract 90 from 126.78 to get the answer of $\frac{5}{2}(cos(36.78)+isin(36.78))$

20. anonymous

now the snap way: $\frac{-3+4i}{(-1-i)^2}=\frac{-3+4i}{2i}$ $=\frac{-3+4i}{2i}\times \frac{-i}{-i}=\frac{4+3i}{2}=2+\frac{3}{2}i$\]

21. anonymous