Question

Suppose w=a+bi is one of the 31st roots of 1.

a. What is the maximum value of a?
b. What is the minimum value of b?

Please help! What does this even mean?!? >.<

Answer 1

@ganeshie8 @mathslover anyone?

Answer 2

Do you know how to find the complex roots of a number, to begin with?

Answer 3

Like using De Moivre's Theorem?

Answer 4

finding max value of a+bi is easy :

since "1" itself is one of the 31st roots of 1, max value for a = 1

Answer 5

you can find the min value for "b" using De Moivre's thm :

31st roots = \(\large e^{i \frac{2\pi k}{31}}\)

Answer 6

\(k = 0, 1, ... 30\)

Answer 7

test those two points ^

Answer 8

31 * 3/4 = 23.25

so the min value occurs at k = 23 or 24

Answer 9

Hmmm I understand the max value part, but my answer key says the min value should be at \[\sin(16\pi/31)\] or about 0.9987

Answer 10

\(\large e^{i \frac{2\pi *23}{31}}\) is clearly one of the 31st roots of 1,

\(\large e^{i \frac{2\pi *23}{31}} = \cos (\frac{46 \pi}{31}) + i \sin (\frac{46 \pi}{31})\)


\(\implies b = \sin (\frac{46 \pi}{31})\)

Answer 11

\(\sin (\frac{46 \pi}{31}) < \sin (\frac{16 \pi}{31})\)

so the min value is \(\sin (\frac{46 \pi}{31})\)

Answer 12

^^thats for minimum value of \(b\) , the imaginary part

Answer 13

i think your text book is talking about min value of \(a\) ?

Answer 14

Oh my fault, it was asking for the MAXIMUM of b, not the minimum. Sorry :S

Answer 15

oh ok then it makes sense :)

Answer 16

maximum for imaginary part occurs at TOP, 1/4th the way to full revolution :

k = 31*1/4 = 7.75

Answer 17

so, test k = 7 and k = 8 points