Jika x=sin 3θ +sin θ dan y=cos 3θ +cos θ, maka nilai x2−y2=...

Pertanyaan

Jika $\mathrm x=\sin\;3\mathrm\theta\;+\sin\;\mathrm\theta\;\mathrm{dan}\;\mathrm y=\cos\;3\mathrm\theta\;+\cos\;\mathrm\theta$ , maka nilai $\mathrm x^2-\mathrm y^2=$ ...

$\cos\;2\mathrm\theta$
$1+\begin{array}{rcl}&&\cos\end{array}\begin{array}{rcl}&&\;\end{array}\begin{array}{rcl}&&2\end{array}\begin{array}{rcl}&&\mathrm\theta\end{array}$
$2-\begin{array}{rcl}&&\cos\end{array}\begin{array}{rcl}&&\;\end{array}\begin{array}{rcl}&&2\end{array}\begin{array}{rcl}&&\mathrm\theta\end{array}$
$\begin{array}{rcl}&&2+2\;\cos\;2\mathrm\theta\end{array}$
$\begin{array}{rcl}&&\sin\;2\mathrm\theta+\cos\;2\mathrm\theta\end{array}$

I. Rachmawati

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah D.

Pembahasan

Ingat kembali:

rumus rumus perbandingan trigonometri untuk selisih dua sudut: $\cos\;(\mathrm A-\mathrm B)=\begin{array}{rcl}&&\sin\end{array}\begin{array}{rcl}&&\;\end{array}\begin{array}{rcl}&&\mathrm A\end{array}\begin{array}{rcl}&&\;\end{array}\begin{array}{rcl}&&\sin\end{array}\begin{array}{rcl}&&\;\end{array}\begin{array}{rcl}&&\mathrm B\end{array}\begin{array}{rcl}&&+\end{array}\begin{array}{rcl}&&\cos\end{array}\begin{array}{rcl}&&\;\end{array}\begin{array}{rcl}&&\mathrm A\end{array}\begin{array}{rcl}&&\;\end{array}\begin{array}{rcl}&&\cos\end{array}\begin{array}{rcl}&&\;\end{array}\begin{array}{rcl}&&\mathrm B\end{array}$ .
identitas: $\sin^2\;\mathrm A+\cos^2\;\mathrm A=1$

Diketahui $\mathrm x=\sin\;3\mathrm\theta\;+\sin\;\mathrm\theta\;\mathrm{dan}\;\mathrm y=\cos\;3\mathrm\theta\;+\cos\;\mathrm\theta$ , maka penyelesaian soal di atas adalah

$\begin{array}{rcl}&&\mathrm x^2+\mathrm y^2\\&=&\left(\sin\;3\mathrm\theta\;+\sin\;\mathrm\theta\right)^2\;+\left(\cos\;3\mathrm\theta\;+\cos\;\mathrm\theta\right)^2\\&=&{\sin^{2\;}3\mathrm\theta\;+\;2\;\sin\;3\mathrm\theta\;\sin\;\mathrm\theta\;+\sin^2\;\mathrm\theta\;+(\cos^2\;3\mathrm\theta\;+2\;\cos\;3\mathrm\theta\;\cos\;\mathrm\theta+\cos^2\;\mathrm\theta)}\\&=&\left(\sin^{2\;}3\mathrm\theta+\cos^2\;3\mathrm\theta\right)+\left(\sin^2\;\mathrm\theta\;+\cos^2\;\mathrm\theta\right)+2\left(\sin\;3\mathrm\theta\;\sin\;\mathrm\theta+\cos\;3\mathrm\theta\;\cos\;\mathrm\theta\right)\\&=&{1+1+2\;\cos\;(3\mathrm\theta-\mathrm\theta)}\\&=&2+2\;\cos\;2\mathrm\theta\end{array}$

Jadi, $\begin{array}{rcl}\mathrm x^2+\mathrm y^2&=&2+2\;\cos\;2\mathrm\theta\end{array}$ .

Oleh karena itu, jawaban yang tepat adalah D.