Thread: Complex No question 1

Hi I need help for the following question,

1.) Let Z1 = 2cis(pi/6) and z2 = rcis(delta) where r>0, and 0<=delta<2pi.
FInd the range of values of r and delta for which z1z2 is
a.) a real number greater than 5
For the modulus of z1z2, I got r>5/2, which was right.
However for the delta, I'm not sure how I'm supposed to do it, it is (pi/6) + delta >5 and then solve the inequality? Seems wrong..
b.) a purely imaginary number with modulus less than 1
How do I find the imaginary number?

Thank you so much for your time, I really appreciate it!
J.

Originally Posted by Tutu

1.) Let Z1 = 2cis(pi/6) and z2 = rcis(delta) where r>0, and 0<=delta<2pi. FInd the range of values of r and delta for which z1z2 is
a.) a real number greater than 5
For the modulus of z1z2, I got r>5/2, which was right.
However for the delta,
b.) a purely imaginary number with modulus less than 1
How do I find the imaginary number?

For to be a real number greater than 5 you must have
.

But also

b) For to be purely imaginary number you must have
be an odd multiple of .

Thanks! May I know how you arrived at pi/6 + thetha = 2kpi?

Why 2kpi, to be exact.

Thank you so much!

Originally Posted by Tutu

Thanks! May I know how you arrived at pi/6 + thetha = 2kpi? Why 2kpi, to be exact.

Suppose that where is a positive real number if and only if .

Is this a rule, thank you!

Originally Posted by Tutu

Is this a rule, thank you!

I don't know what you mean by rule.

If and it is necessary that .

If it is necessary that .
Thus .

But in the question, we want a positive real number.

But why is it that when you want a positive real number, it has to be 2kpi?
I think I kind of understand..
Does that meant that the angle has to be in the positive direction to get a positive real number hence 2kpi, not -2kpi or anything?

Originally Posted by Tutu

But why is it that when you want a positive real number, it has to be 2kpi?
I think I kind of understand..
Does that meant that the angle has to be in the positive direction to get a positive real number hence 2kpi, not -2kpi or anything?

so it must be an even multiple if .

So in , the integer can be positive or negative.

Recall .

Don't forget that you need

How did you get
cos((2k+1)pi) = -1?

More specifically, why 2K+1 and how did you know the whole eqn equals to -1?

Thank you very much once again!