# Trouble with complex numbers

• May 7th 2012, 12:25 PM
maybealways
Trouble with complex numbers
The question is this:

$\displaystyle |(2+i)(cos\frac{\pi}{6}-i sin\frac{\pi}{6})|$

I'm not sure if what I am doing is right...

$\displaystyle |(2+i)(\frac{\sqrt3}{2}-\frac{i}{2})|$

But I'm not sure what to do after. Is it just simple algebra? Because if it is, I did something really weird and didn't get the answer, which is supposed to be $\displaystyle \sqrt5$

Any help would be wonderful! Any working would be great, to see where I went wrong.
• May 7th 2012, 01:18 PM
hairymclairy
Re: Trouble with complex numbers
You're doing fine, just multiply out the brackets and write it as $\displaystyle x+yi$ for some values x and y. Then in order to find the absolute value, you simply take the square root of $\displaystyle x^2+y^2$. If you picture the complex number $\displaystyle x+yi$ on the complex plane, where the x axis corresponds to the real part and the y axis corresponds to the imaginary part, then $\displaystyle |x+yi|$ is simply the hypotenuse of a right angle triangle with horizontal side of length $\displaystyle x$ and vertical side of length $\displaystyle y$.
• May 7th 2012, 01:59 PM
Plato
Re: Trouble with complex numbers
Quote:

Originally Posted by maybealways
The question is this:
$\displaystyle |(2+i)(cos\frac{\pi}{6}-i sin\frac{\pi}{6})|$
I'm not sure if what I am doing is right...
$\displaystyle |(2+i)(\frac{\sqrt3}{2}-\frac{i}{2})|$
You can really simplify things by noting $\displaystyle |z\cdot w|=|z|\cdot |w|.$
Thus $\displaystyle |(2+i)(\cos\frac{\pi}{6}-i \sin\frac{\pi}{6})|=|(2+i)|\cdot|(\cos\frac{\pi}{6 }-i \sin\frac{\pi}{6})|$
Moreover $\displaystyle |\cos(\theta)\pm i \sin(\theta)|=1,~\forall\theta\in\mathbb{R}~.$