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\textbf{Trigonometric Equations}

\textbf{Ex 1} Solve the equation $\sin^2 \theta +\sin \theta =0$ for 
$0\le \theta <2\pi$.

\textbf{Sol} $\sin^2 \theta +\sin \theta =0 \Rightarrow$
$\sin \theta (\sin \theta +1)=0 \Rightarrow $
a) $\sin \theta =0$ or b) $\sin \theta =-1$ , so
a) $\theta=0$ or $\theta=\pi$ or b) $\theta =3\pi /2$.

\textbf{Ex 2} Find all solutions of the equation $\cos^2 \theta =1$.

\textbf{Sol} $\cos^2 \theta =1$ gives $\cos \theta = \pm 1$; and
$\cos \theta = 1$ for $\theta = 2n\pi $  (where $n$ is any integer) and

$\cos \theta =-1$ for $\theta = (2n+1)\pi$ (where $n$ is any integer).

Combining these solutions gives that $\theta=k\pi$, where k is any integer.

\textbf{Pr A} Find all solutions of the equation $\sin\theta=\cos\theta$.

\textbf{Pr B} Solve the equation $\sin\theta=\sqrt{3}\cos\theta$ for
$0\le \theta <2\pi$.

\textbf{Pr C} Solve the equation $\sin 2\theta=\cos\theta$ for
$0\le \theta <2\pi$.

\textbf{Pr 1} Solve the equation $2\sin^2 \theta - \sin \theta =0$ for
	       $0\le \theta < 2\pi$.
	       
\textbf{Pr 2} Solve the equation $\cos^2 \theta +\cos \theta =0$ for
	      $0\le \theta < 2\pi$.
	      
\textbf{Pr 3} Solve the equation $\cos \theta - \sin \theta =1$ for	       
		$0\le \theta < 2\pi$.
		
\textbf{Pr 4} Find all solutions of the equation $\tan^2 \theta =\tan \theta$.

\textbf{Pr 5} Solve the equation $2\sin^2 \theta -\cos \theta -1=0$ for
              $0\le \theta < 2\pi$.

\textbf{Pr 6} Solve the equation $\cos 2\theta = \sin \theta$ for
		$0\le \theta < 2\pi$.
		
\textbf{Pr 7} Solve the equation $2\cos^2 3\theta = \cos 3\theta$ for
		$0\le \theta < 2\pi$.
		
\textbf{Pr 8} Solve the equation $\cos 4\theta=2-3\sin 2\theta$ for
             $0\le \theta <2\pi$.		
		
		
		


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