Question

Solve sin x - (3sin x-1) = 0

Answer 1

`\large\boxed{x=(\pi)/(6)+2k\pi\ \vee\ x=(5\pi)/(6)+2k\pi,\ k\in\mathbb{Z}}`

Explanation:

`\sin x-(3\sin x-1)=0\\\\\sin x-3\sin x+1=0\qquad\text{subtract 1 from both sides}\\\\-2\sin x=-1\qquad\text{divide both sides by 2}\\\\\sin x=(1)/(2)\Rightarrow x=(\pi)/(6)+2k\pi\ \vee\ x=(5\pi)/(6)+2k\pi,\ k\in\mathbb{Z}`

`\text{Equation:}\\\\\sin x=a\\\\\text{has solutions}\\\\x=\theta+2k\pi\ \vee\ x=(\pi-\theta)+2k\pi\\\\\text{Why}\ 2k\pi?\\\text{Because the sine function has a period of}\ 2\pi.`

`\text{look at the table}\\\\\sin x=(1)/(2)\to x=(\pi)/(6)\ \vee\ x=\pi-(\pi)/(6)=(5\pi)/(6)`

`\text{Other solution:}\\\\\sin x=(1)/(2)\Rightarrow x=\sin^(-1)(1)/(2)\\\\x=(\pi)/(6)+2k\pi\ \vee\ x=(5\pi)/(6)+2k\pi`