Pembahasan trigonometri (5)


Untuk soal nomor 7
jika \alpha+\beta+\gamma=180^{o}, maka kita dapatkan
\alpha+\beta=180^{o}-\gamma
sehingga \tan(\alpha+\beta)=\tan(180^{o}-\gamma)
\frac{\tan\alpha+\tan\beta}{1-\tan\alpha.\tan\beta}=-\tan\gamma
\tan\alpha+\tan\beta=(1-\tan\alpha.\tan\beta)-\tan\gamma
\tan\alpha+\tan\beta=-\tan\gamma+\tan\alpha.\tan\beta\tan\gamma
\tan\alpha+\tan\beta+\tan\gamma=\tan\alpha.\tan\beta\tan\gamma
Terbukti.

Untuk soal nomor 8.
jika \alpha+\beta+\gamma=180^{o}, maka kita dapatkan
\frac{1}{2}\alpha+\frac{1}{2}\beta+\frac{1}{2}\gamma=90^{o}

dari soal nomor 1 kita telah memperoleh kesamaan bahwa
\cos^{2}a+\cos^{2}b+\cos^{2}c=2(1+\sin a\sin b\sin c) untuk a+b+c=90^{o}.
Sehingga
\cos^{2}\frac{1}{2}\alpha+\cos^{2}\frac{1}{2}\beta+\cos^{2}\frac{1}{2}\gamma= 2(1+\sin\frac{1}{2}\alpha\sin\frac{1}{2}\beta\sin\frac{1}{2}\gamma)
\frac{1+\cos\alpha}{2}+\frac{1+\cos\beta}{2}+\frac{1+\cos\gamma}{2}= 2(1+\sin\frac{1}{2}\alpha\sin\frac{1}{2}\beta\sin\frac{1}{2}\gamma)
1+\cos\alpha+1+\cos\beta+1+\cos\gamma=4(1+\sin\frac{1}{2}\alpha\sin\frac{1}{2}\beta\sin\frac{1}{2}\gamma)
\cos\alpha+\cos\beta+\cos\gamma=1+4\sin\frac{1}{2}\alpha\sin\frac{1}{2}\beta\sin\frac{1}{2}\gamma
Terbukti.

Untuk soal nomor 5
jika \alpha+\beta+\gamma=45^{o}, maka kita dapatkan
\cos(\alpha+\beta)=\cos(45^{o}-\gamma), \cos(\alpha+\gamma)=\cos(45^{o}-\beta) dan
\cos(\beta+\gamma)=\cos(45^{o}-\alpha), sehingga
\cos(45^{o}-\alpha)\cos(45^{o}-\beta)\cos(45^{o}-\gamma)=
\cos(\beta+\gamma)\cos(\alpha+\gamma)\cos(\alpha+\beta)
= \frac{1}{2}(\cos(\alpha+\beta+2\gamma)+\cos(\alpha-\beta))\cos(\alpha+\beta)
= \frac{1}{2}\cos(\alpha+\beta+2\gamma)\cos(\alpha+\beta)+\frac{1}{2}\cos(\alpha-\beta)\cos(\alpha+\beta)
=\frac{1}{2}(\frac{1}{2}(\cos(2\alpha+2\beta+2\gamma)+\cos 2\gamma))+\frac{1}{2}(\frac{1}{2}\cos(2\alpha)+\cos(2(-\beta))
= \frac{1}{4}(\cos(90)^{o}+\cos 2\gamma)+\frac{1}{4}(\cos(2\alpha)+\cos(2\beta))
= \frac{1}{4}\cos (2\gamma)+\frac{1}{4}\cos(2\alpha)+\frac{1}{4}\cos(2\beta)
dengan mengalikan kedua ruas dengan 4 kita dapatkan:
4(\cos(45^{o}-\alpha)\cos(45^{o}-\beta)\cos(45^{o}-\gamma)=
\cos (2\gamma)+\cos(2\alpha)+\cos(2\beta)

terbukti.

Tinggalkan Balasan

Isikan data di bawah atau klik salah satu ikon untuk log in:

Logo WordPress.com

You are commenting using your WordPress.com account. Logout / Ubah )

Gambar Twitter

You are commenting using your Twitter account. Logout / Ubah )

Foto Facebook

You are commenting using your Facebook account. Logout / Ubah )

Foto Google+

You are commenting using your Google+ account. Logout / Ubah )

Connecting to %s