# Math Help - Complex Number

1. ## Complex Number

Simplify without the use of calculator

[cos (pi/7) – i sin (pi/7)]^3 / [ cos (pi/7) + i sin (pi/7)]^4

How can I solve this?

2. Hello geton,

From your posts elsewhere I think you know the $Re^{i\theta}$ form of a complex number.

In this case you wont need the R.

Once you have them in that form it should be easy to simplify.

3. Originally Posted by a tutor
Hello geton,

From your posts elsewhere I think you know the $Re^{i\theta}$ form of a complex number.

In this case you wont need the R.

Once you have them in that form it should be easy to simplify.
I know z₁ = r(cosѲ + i sinѲ) & z₂ = r(cosѲ - i sinѲ). But how can I solve by this?

4. Recall this basic principle:
$\left[ {\cos \left( {\frac{\pi }{7}} \right) - i\sin \left( {\frac{\pi }{7}} \right)} \right]^3 = \left[ {\cos \left( {\frac{{ - \pi }}{7}} \right) + i\sin \left( {\frac{{ - \pi }}{7}} \right)} \right]^3$ $= \left[ {\cos \left( {\frac{{ - 3\pi }}{7}} \right) + i\sin \left( {\frac{{ - 3\pi }}{7}} \right)} \right]$.

Thus:
$\frac{{\left[ {\cos \left( {\frac{\pi }{7}} \right) - i\sin \left( {\frac{\pi }{7}} \right)} \right]^3 }}{{\left[ {\cos \left( {\frac{\pi }{7}} \right) + i\sin \left( {\frac{\pi }{7}} \right)} \right]^4 }} = \left[ {\cos \left( { - \pi } \right) + i\sin \left( { - \pi } \right)} \right]$

5. Originally Posted by Plato
Recall this basic principle:
$\left[ {\cos \left( {\frac{\pi }{7}} \right) - i\sin \left( {\frac{\pi }{7}} \right)} \right]^3 = \left[ {\cos \left( {\frac{{ - \pi }}{7}} \right) + i\sin \left( {\frac{{ - \pi }}{7}} \right)} \right]^3$ $= \left[ {\cos \left( {\frac{{ - 3\pi }}{7}} \right) + i\sin \left( {\frac{{ - 3\pi }}{7}} \right)} \right]$.

Thus:
$\frac{{\left[ {\cos \left( {\frac{\pi }{7}} \right) - i\sin \left( {\frac{\pi }{7}} \right)} \right]^3 }}{{\left[ {\cos \left( {\frac{\pi }{7}} \right) + i\sin \left( {\frac{\pi }{7}} \right)} \right]^4 }} = \left[ {\cos \left( { - \pi } \right) + i\sin \left( { - \pi } \right)} \right]$

But what about denominator. I didn't understand properly. Can you explain for me?

6. Originally Posted by geton
But what about denominator. I didn't understand properly. Can you explain for me?
Using deMoivre's Theorem:

$\left[ {\cos \left( {\frac{\pi }{7}} \right) + i\sin \left( {\frac{\pi }{7}} \right)} \right]^4 = {\cos \left( {\frac{{4\pi }}{7}} \right) + i\sin \left( {\frac{{4\pi }}{7}} \right)}$

Then you just divide numerator by denominator using the usual rule for division of complex numbers expressed in polar form to get:

$\cos \left( -\frac{3 \pi}{7} - \frac{4 \pi}{7} \right) + i \sin \left( -\frac{3\pi}{7} - \frac{4\pi}{7} \right) = ...$