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Problem - Tangen sudut rangkap | Catatan Nurkholis

Problem – Tangen sudut rangkap

Problem:
Buktikan bahwa $$begin{aligned}tan ntheta &=frac{displaystylebinom{n}{1} tan theta – binom{n}{3} tan^3theta+cdots}{displaystyle1-binom{n}{2} tan^2theta+cdots}end{aligned}$$

Solusi:

Dengan menggunakan De Moivre’s Theorem $e^{(itheta)n}=e^{i(ntheta)}=cosleft(nthetaright)+isinleft(nthetaright)$. Sementara itu, $e^{(itheta)n}=left(costheta+isinthetaright)^n$. Sehingga bisa kita tuliskan
$$
begin{aligned}
cosleft(nthetaright)+isinleft(nthetaright)&=sum_{k=0}^nbinom{n}{k}i^kcos^{n-k}thetasin^ktheta
end{aligned}
$$
Tetapi kita tahu bahwa $i^2=-1$ dan $i^3=-i$. Sehingga bentuk di atas dapat disederhanakan menjadi

$$
begin{aligned}
cosleft(nthetaright)+isinleft(nthetaright)&=sum_{k=0}^nbinom{n}{2k}(-1)^kcos^{n-2k}thetasin^{2k}theta+sum_{k=0}^nbinom{n}{2k+1}(i)^{2k+1}cos^{n-2k-1}thetasin^{2k+1}theta
end{aligned}
$$
Dengan memisahkan antara nilai Real dan Imaginer pada bentuk di atas, diperoleh:
$$cos(ntheta)=sum_{k=0}^nbinom{n}{2k}(-1)^kcos^{n-2k}thetasin^{2k}theta$$
$$sin(ntheta)=sum_{k=0}^nbinom{n}{2k+1}(-1)^kcos^{n-2k-1}thetasin^{2k+1}theta$$
Selanjutnya, kita tahu bahwa $displaystyletan(ntheta)=frac{sin(ntheta)}{cos(ntheta)}$. Sehingga bentuk di atas dapat diubah menjadi
$$
begin{aligned}
tan(ntheta)&=frac{displaystylesum_{k=0}^nbinom{n}{2k+1}(-1)^kcos^{n-2k-1}thetasin^{2k+1}theta}{displaystylesum_{k=0}^nbinom{n}{2k}(-1)^kcos^{n-2k}thetasin^{2k}theta}
end{aligned}
$$
Bagi pembilang dan penyebut dengan $cos^ntheta$ diperoleh

$$
begin{aligned}
tan(ntheta)&=frac{displaystylesum_{k=0}^nbinom{n}{2k+1}(-1)^ktan^{2k+1}theta}{displaystylesum_{k=0}^nbinom{n}{2k}(-1)^ktan^{2k}theta}\
&=frac{displaystylebinom{n}{1} tan theta – binom{n}{3} tan^3theta+cdots}{displaystyle1-binom{n}{2} tan^2theta+cdots}
end{aligned}
$$

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