Gjest skrev:Vis at:
[tex]\cos(56^{\circ})*\cos\left (2*56^{\circ} \right )*\cos(2^2*56^{\circ})*...*\cos(2^{23}*56^{\circ})=\frac{1}{2^{24}}[/tex]
Forslag.
Innfører:
[tex]\displaystyle\text{C}=\prod_{r=1}^{24}\cos{\left(2^{r-1}*56^o\right)}[/tex]
[tex]\displaystyle\text{S}=\prod_{r=1}^{24}\sin{\left(2^{r-1}*56^o\right)}[/tex]
så:
[tex]C*S = \left(\sin(56^o)*\cos(56^o)\right)*\left(\sin(2*56^o)*\cos(2*56^o)\right)*...*\left(\sin(2^{23}*56^o)*\cos(2^{23}*56^o)\right)[/tex]
videre:
[tex]C*S = \frac{1}{2^{24}}\left(2\sin(56^o)*\cos(56^o)\right)*\left(2\sin(2*56^o)*\cos(2*56^o)\right)*...*\left(2\sin(2^{23}*56^o)*\cos(2^{23}*56^o)\right)[/tex]
fordi:
[tex]\sin(2x)=2\sin(x)\cos(x)[/tex]
altså:
[tex]C*S = \frac{1}{2^{24}}\left(\sin(2*56^o)*\sin(2^2*56^o)*...*\sin(2^{24}*56^o)\right)[/tex]
der
[tex]\sin(x) = \sin(180^o - x)[/tex]
[tex]=>[/tex]
[tex]C*S = \frac{1}{2^{24}}\left(\sin(56^o)*\sin(2*56^o)*...*\sin(2^{23}*56^o)\right)[/tex]
[tex]=>[/tex]
[tex]C*S = \frac{1}{2^{24}} * S[/tex]
der
[tex]C = \frac{1}{2^{24}}[/tex]
for
[tex]S \neq 0[/tex]