Associate angles | Example-1

Associate angles: Example

Solve in the interval \(0^{o}\leq \theta \leq 360^{o}\) the equation \(\sqrt{3}\sin\theta+\cos\theta=1\)

Solve:
\(\sqrt{3}\sin\theta+\cos\theta\)=1
or, \(\sqrt{3}\sin\theta=1-\cos\theta\)

Squaring both side we get
or, \((\sqrt{3}\sin\theta)^{2}\)=\((1-\cos\theta)^{2}\)
or, \(3(1-\cos^{2}\theta)\)=1-\(2\cos\theta+\cos^{2}\theta\)
or, \(4\cos^{2}\theta-2\cos\theta-2\)=0
or, \(2\cos^{2}\theta-\cos\theta-1\)=0
or, \((\cos\theta-1)(2\cos\theta+1)\)=0
Either \(\cos\theta=1\)

\(\theta=0^{o}\) (for first quardent)
\(\theta=360^{o}\) (for forth quardent)
or, \(\cos\theta=-\frac{1}{2}\)
\(\theta=180^{o}-60^{o} =120^{o}\)(for second quardent)
\(\theta=180^{o}+60^{o} =240^{o}\) (for third quardent)

when \(\theta=240^{o}\) it can't satisfy the equation. So \(\theta=0^{o}, 120^{o}, 360^{o}\)