Question

The number of solutions of the equation $\sin \theta +\cos \theta =\sin 2\theta$ in the interval $[-\pi ,\pi ]$ is?

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is B $2$

$\sin \theta +\cos \theta =\sin 2\theta =(\sin \theta +\cos \theta {)}^2-1$

$\Rightarrow (sin\theta +\cos \theta {)}^2-(\sin \theta +\cos \theta )-1=0$

$\therefore sin\theta +\cos \theta =\frac{1-\sqrt{5}}{2}$

$\Rightarrow \cos (\pi /4-\theta )=\frac{1-\sqrt{5}}{2}=-\frac{(\sqrt{5}-1)}{2}$

Thus, there are two possible solutions for $\theta$ between $-\pi$ to $\pi$.