Trigonometric Equations

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

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) $\theta =3\pi /2$.

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

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.

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

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

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

Pr 1 Solve the equation $2\sin^2 \theta - \sin \theta =0$ for $0\le \theta <2\pi$.

Pr 2 Solve the equation $\cos^2 \theta +\cos \theta =0$ for $0\le \theta <2\pi$.

Pr 3 Solve the equation $\cos \theta - \sin \theta =1$ for $0\le \theta <2\pi$.

Pr 4 Find all solutions of the equation $\tan^2 \theta =\tan \theta$.

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

Pr 6 Solve the equation $\cos 2\theta = \sin \theta$ for $0\le \theta <2\pi$.

Pr 7 Solve the equation $2\cos^2 3\theta = \cos 3\theta$ for $0\le \theta <2\pi$.

Pr 8 Solve the equation $\cos 4\theta=2-3\sin 2\theta$ for $0\le \theta <2\pi$.

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