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?
Recall this basic principle:
$\displaystyle \left[ {\cos \left( {\frac{\pi }{7}} \right) - i\sin \left( {\frac{\pi }{7}} \right)} \right]^3 = \left[ {\cos \left( {\frac{{ - \pi }}{7}} \right) + i\sin \left( {\frac{{ - \pi }}{7}} \right)} \right]^3$ $\displaystyle = \left[ {\cos \left( {\frac{{ - 3\pi }}{7}} \right) + i\sin \left( {\frac{{ - 3\pi }}{7}} \right)} \right]$.
Thus:
$\displaystyle \frac{{\left[ {\cos \left( {\frac{\pi }{7}} \right) - i\sin \left( {\frac{\pi }{7}} \right)} \right]^3 }}{{\left[ {\cos \left( {\frac{\pi }{7}} \right) + i\sin \left( {\frac{\pi }{7}} \right)} \right]^4 }} = \left[ {\cos \left( { - \pi } \right) + i\sin \left( { - \pi } \right)} \right]$
Using deMoivre's Theorem:
$\displaystyle \left[ {\cos \left( {\frac{\pi }{7}} \right) + i\sin \left( {\frac{\pi }{7}} \right)} \right]^4 = {\cos \left( {\frac{{4\pi }}{7}} \right) + i\sin \left( {\frac{{4\pi }}{7}} \right)}$
Then you just divide numerator by denominator using the usual rule for division of complex numbers expressed in polar form to get:
$\displaystyle \cos \left( -\frac{3 \pi}{7} - \frac{4 \pi}{7} \right) + i \sin \left( -\frac{3\pi}{7} - \frac{4\pi}{7} \right) = ...$