Jawab:
[tex]\large\text{$\begin{aligned}&x=\frac{\pi}{3},\ x=\frac{5\pi}{3}\\end{aligned}$}[/tex]x = π/3, x = 5π/3
Penjelasan dengan langkah-langkah:
[tex]\large\text{$\begin{aligned}&-2\sin x-3\cot x+3\csc x=0\ \:\textsf{dengan}\:\ \frac{\pi}{4}<x<2\pi\\\\&-2\sin x-3\left(\frac{\cos x}{\sin x}\right )+3\left(\frac{1}{\sin x}\right)=0\\\\&\frac{-2\sin^2x-3\cos x+3}{\sin x}=0\\\\&\frac{-2\left(1-\cos^2x\right)-3\cos x+3}{\sin x}=0\\\\&\frac{-2+2\cos^2x-3\cos x+3}{\sin x}=0\\\\&\frac{2\cos^2x-3\cos x+1}{\sin x}=0\\\\&2\cos^2x-3\cos x+1=0\ \:\textsf{dengan}\:\ \sin x\neq0\end{aligned}$}[/tex]
[tex]\large\text{$\begin{aligned}&\textsf{Pemfaktoran sesuai bentuk }\normalsize\text{$2a^2-3a+1=(2a-1)(a-1)$}:\\&(2\cos x-1)(\cos x-1)=0\end{aligned}$}[/tex]
[tex]\large\text{$\begin{aligned}&\begin{array}{ll}\bullet\ &2\cos x-1=0\implies2\cos x=1\\&\cos x=\frac{1}{2}\quad\textsf{dengan}\:\ \frac{\pi}{4}<x<2\pi\\\\&\boxed{x=\frac{\pi}{3},\ x=\frac{5\pi}{3}}\\\\\bullet\ &\cos x-1=0\implies\cos x=1\\&\textsf{Dengan$\:\ \frac{\pi}{4}<x<2\pi:\ $tidak ada solusi.}\\&\textsf{Walaupun rentang nilai $x$ diperbesar,}\\&\textsf{tetap tidak ada solusi dari kemungkinan ini}\\&\textsf{karena ketika $|\cos x|=1$, $\sin x=0$.}\end{array}\end{aligned}$}[/tex]
∴ Maka, nilai x yang memenuhi persamaan
–2 sin x – 3 cot x + 3 csc x = 0 dengan π/4 < x < 2π adalah:
[tex]\large\text{$\begin{aligned}&\boxed{\ x=\frac{\pi}{3},\ x=\frac{5\pi}{3}\ }\end{aligned}$}[/tex]
[answer.2.content]
