Thread: Complex Numbers Basics!

Hi I'm confused with these three questions. I do not have the answers with me so I really need your help!

1.) If z = 4cis40, find -z.
I know z is a+bi since 4cis40 is in the first quadrant, so -z is -a-bi, which will be in the third quadrant.
So I got an answer of 4cis(-220) since I was told that the argument in third and fourth quadrants are always negative but the answer is 4cis220!
Can someone help me with this?

2.) 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?

3.) Let z=cis(a)
a.) Use De Moivre's theorem to show that z^n + (1/z)^n = 2cos(na)
I was able to prove this part.
b.) Use the binomial theorem to expand (z + (1/z))^4
I got z^4 + 4z^2 +6 +(4/z^2) + (1/z^4), is it right?
Part 2, Hence show that cos^4(a) = (1/8) (cos(4a) + 4cos(2a) + 3 and find the integral of cos^4(a)da.
I'm not sure how to show..


Thank you so so much,
J.

Hey Tutu.

The third quadrant lies in the [180,270) range which means 220 will lie in the third quadrant.

Remember that the angle is measured in the positively orientated counter-clockwise orientation not in the negative clockwise direction.

Hi thank you! So is it that 4cis(220) = 4cis(-140)?

Yeah that's right.

Thank you so much!