Hitung Nilai Berikut Cos 2Phi Per 3 - Cos 5Phi Per

Berikut ini adalah pertanyaan dari anyra142538 pada mata pelajaran Matematika untuk jenjang Sekolah Menengah Atas

Hitung Nilai BerikutCos 2Phi Per 3 - Cos 5Phi Per 3 + Sin 2Phi Per 3

*Disertai Cara Yaaa
Ngasal ? Report

Jawaban dan Penjelasan

Berikut ini adalah pilihan jawaban terbaik dari pertanyaan diatas.

$\displaystyle\cos\left(\frac{2\pi}{3}\right)-\cos\left(\frac{5\pi}{3}\right)+\sin\left(\frac{2\pi}{3}\right)$

$\large\text{$\begin{aligned}&=\bf-1+\frac{\sqrt{3}}{2}\end{aligned}$}$

Pembahasan

Trigonometri

$\begin{aligned}&\cos\left(\frac{2\pi}{3}\right)-\cos\left(\frac{5\pi}{3}\right)+\sin\left(\frac{2\pi}{3}\right)\\\\&\quad\left[\ \begin{aligned}\cos\alpha-\cos\beta=-2\sin\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)\end{aligned}\right.\\\\&{=\ }-2\sin\left(\frac{\frac{2\pi}{3}+\frac{5\pi}{3}}{2}\right)\sin\left(\frac{\frac{2\pi}{3}-\frac{5\pi}{3}}{2}\right)+\sin\left(\frac{2\pi}{3}\right)\end{aligned}$
$\begin{aligned}&{=\ }-2\sin\left(\frac{7\pi}{6}\right)\sin\left(-\frac{3\pi}{6}\right)+\sin\left(\frac{2\pi}{3}\right)\\&{=\ }2\sin\left(\frac{7\pi}{6}\right)\sin\left(\frac{3\pi}{6}\right)+\sin\left(\frac{2\pi}{3}\right)\end{aligned}$
$\begin{aligned}&{=\ }2\sin\left(\frac{7\pi}{6}\right)\sin\left(\frac{\pi}{2}\right)+\sin\left(\frac{2\pi}{3}\right)\\\\&\quad\left[\ \begin{aligned}\sin\left(\frac{7\pi}{6}\right)&=\sin\left(\pi+\frac{\pi}{6}\right)\\&=-\sin\left(\frac{\pi}{6}\right)\\&=-\frac{1}{2}\\\sin\left(\frac{\pi}{2}\right)&=1\\\end{aligned}\right.\\\\&{=\ }\cancel{2}\cdot\left(-\frac{1}{\cancel{2}}\right)\cdot1+\sin\left(\frac{2\pi}{3}\right)\\&{=\ }-1+\sin\left(\frac{2\pi}{3}\right)\end{aligned}$
$\begin{aligned}&{=\ }-1+\sin\left(\frac{4\pi}{6}\right)\\&{=\ }-1+\sin\left(\frac{\pi}{2}+\frac{\pi}{6}\right)\\\\&\quad\left[\ \begin{aligned}\sin\left(\frac{\pi}{2}+\alpha\right)&=\cos\alpha\end{aligned}\right.\\\\&{=\ }-1+\cos\left(\frac{\pi}{6}\right)\\&{=\ }\boxed{\ \bf-1+\frac{\sqrt{3}}{2}\ }\end{aligned}$