Question

question 14. (5 marks) find all the solutions of the followi 2\cos(x)\sin(x) - \sin(x) = 0

Explanation:

Step1: Factor out common term

Factor out \(\sin(x)\) from the left - hand side of the equation \(2\cos(x)\sin(x)-\sin(x) = 0\). We get \(\sin(x)(2\cos(x)-1)=0\).

Step2: Solve for each factor

According to the zero - product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\).

• Case 1: \(\sin(x)=0\)

The general solution for the equation \(\sin(x)=0\) is \(x = n\pi\), where \(n\in\mathbb{Z}\) (the set of all integers).

• Case 2: \(2\cos(x)-1 = 0\)

First, solve the equation \(2\cos(x)-1=0\) for \(\cos(x)\). Add 1 to both sides: \(2\cos(x)=1\), then divide both sides by 2: \(\cos(x)=\frac{1}{2}\).
The general solution for the equation \(\cos(x)=\frac{1}{2}\) is \(x = 2n\pi\pm\frac{\pi}{3}\), where \(n\in\mathbb{Z}\).

Answer:

The solutions of the equation \(2\cos(x)\sin(x)-\sin(x) = 0\) are \(x=n\pi\) or \(x = 2n\pi\pm\frac{\pi}{3}\), where \(n\in\mathbb{Z}\)