Før nyttårsnøtt

[introad]

Hvis [tex]\left [ 2\left (cos\frac{\pi }{12}+isin\frac{\pi}{12} \right ) \right ]^{5}=a+bi[/tex]
hvor [tex]a\cup b\in\mathbb{R}[/tex]
Finn [tex]a+b[/tex]

[Stringselings]

Fra Eulersformel. [tex](2 \ (cos \ \pi/12+i\sin\pi/12 \ ))^5=(2e^{i \pi/12})^5=32e^{i \cdot 5\pi/12}=32cos \ 5\pi/12+32i\sin5\pi/12=a+bi[/tex]

[tex]a+b=32(cos \ 5\pi/12+sin \ 5\pi/12)=32 \sqrt 2 \ sin \ 2\pi/3=32 \sqrt 2 \cdot \sqrt 3/2=16\sqrt6[/tex]