Complex Numbers Problems

[Spectral]

Hey, I had a small math test in class on the subject of complex numbers, on which I seem to have gotten my *** handed to me. Specifically, I could use some help with two fairly simple-looking problems.


1) For any complex z and w, interpret the expression zw* + z*w

I took general forms of z = a + bi and w = c + di and expanded the problem to end up with 2(ac + bd). From there I'm not sure how to proceed.




For this one, I'm not sure how to insert a theta so please imagine 'Q' as one.

2) Express (cosQ + isinQ) / (cos3Q - isin3Q) in the form a + ib, where a and b are real numbers.

I'm not very sure where to start with this one.



Thanks and let me know if I can make anything clearer.

 

[pka]

Hey, I had a small math test in class on the subject of complex numbers, on which I seem to have gotten my *** handed to me. Specifically, I could use some help with two fairly simple-looking problems.
1) For any complex z and w, interpret the expression zw* + z*w

I took general forms of z = a + bi and w = c + di and expanded the problem to end up with 2(ac + bd). From there I'm not sure how to proceed.

2) Express (cosQ + isinQ) / (cos3Q - isin3Q) in the form a + ib, where a and b are real numbers.
I'm not very sure where to start with this one.

Then is it \(\displaystyle \cos^3(\theta)-i\sin^3(\theta)\) OR is it \(\displaystyle \cos(3\theta)-i\sin(3\theta)\)???

 

[Deleted member 4993]

Hey, I had a small math test in class on the subject of complex numbers, on which I seem to have gotten my *** handed to me. Specifically, I could use some help with two fairly simple-looking problems.


1) For any complex z and w, interpret the expression zw* + z*w

I took general forms of z = a + bi and w = c + di and expanded the problem to end up with 2(ac + bd). From there I'm not sure how to proceed.




For this one, I'm not sure how to insert a theta so please imagine 'Q' as one.

2) Express (cosQ + isinQ) / (cos3Q - isin3Q) in the form a + ib, where a and b are real numbers.

I'm not very sure where to start with this one.



Thanks and let me know if I can make anything clearer.

For 2, start with DeMoivre's theorem:

cos(3Θ) - i sin(3Θ) = e^{-i(3Θ + 2kπ)}

 

[Spectral]

In the first it should be \(\displaystyle 2(ac-bd)\) Check your algebra.

Then is it \(\displaystyle \cos^3(\theta)-i\sin^3(\theta)\) OR is it \(\displaystyle \cos(3\theta)-i\sin(3\theta)\)???

Appreciate the reply.

Are you sure? I ended up with 2ac -2bi^2d, which would result in 2ac - 2bd(-1) thus reverting the negativity.

My whole expansion was


ac - adi + bic - bi^2 + ac + adi - bic - bi^2d



For the second question it is the latter, 3 times theta