The Student Room Group

De Moivre's Theorem...

How would I find the modulus and principle argument of this monster?

4+4i(3+i)5\frac{4+4i}{(\sqrt{3}+i)^5}

:hmmmm:
Original post by RamocitoMorales
How would I find the modulus and principle argument of this monster?

4+4i(3+i)5\frac{4+4i}{(\sqrt{3}+i)^5}

:hmmmm:


Work out the modulus and principle argument for the component parts and then add/subtract multiply/divide as necessary.
Original post by ghostwalker
Work out the modulus and principle argument for the component parts and then add/subtract multiply/divide as necessary.


For the top part, I find that the modulus is 424\sqrt{2} and the argument is π4\frac{\pi}{4}. For the bottom part, I find that the modulus is 3232 and the argument is 5π6\frac{5\pi}{6}.

So what do I do? :puppyeyes:

I tried combining the two fractions together, but I ended up with something stupid,

z=4+4i=42(cosπ4+isinπ4)z=4+4i=4\sqrt{2}(cos\frac{\pi}{4}+isin\frac{\pi}{4})

w=(3+i)5=32(cos5π6+isin5π6)w=(\sqrt{3}+i)^5=32(cos\frac{5\pi}{6}+isin\frac{5\pi}{6})

zw=12864i+643i1024\frac{z}{w}=\frac{128-64i+64\sqrt{3}i}{1024}

:cry2:
Original post by RamocitoMorales
For the top part, I find that the modulus is 424\sqrt{2} and the argument is π4\frac{\pi}{4}. For the bottom part, I find that the modulus is 3232 and the argument is 5π6\frac{5\pi}{6}.

So what do I do? :puppyeyes:

I tried combining the two fractions together, but I ended up with something stupid,

z=4+4i=42(cosπ4+isinπ4)z=4+4i=4\sqrt{2}(cos\frac{\pi}{4}+isin\frac{\pi}{4})

w=(3+i)5=32(cos5π6+isin5π6)w=(\sqrt{3}+i)^5=32(cos\frac{5\pi}{6}+isin\frac{5\pi}{6})

zw=12864i+643i1024\frac{z}{w}=\frac{128-64i+64\sqrt{3}i}{1024}

:cry2:


When you divide two complex numbers, one by the other, all you need do is divide their modulii, and subtract their arguments (the right way round of course).
Original post by ghostwalker
When you divide two complex numbers, one by the other, all you need do is divide their modulii, and subtract their arguments (the right way round of course).


It worked!

Original post by RamocitoMorales
It worked!



Your previous answer should work out correctly as well, but I can't be arsed to check it.

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