I tried Wolframalpha to solve this equation. The solution is $\theta\approx 0.355$. Since once a wise guy at MSE told me not to trust machines, I would like to know what methods can be used to solve this equation. I would appreciate for a brief explanation.
By the way $0\leq\theta\leq\pi/4$.
My second question is for any $x>0$ and $x\in\mathbb{R}$, Does $$x\sin 2\theta +\sin \theta =1$$ have a real valued solution for $0\leq\theta\leq\pi/4$?
Best Answer
Here’s a hint: start by using the double-angle formula $$\sin 2\theta = 2\sin \theta \cos \theta,$$ then use the identity $$\sin^2 \theta + \cos^2 \theta = 1.$$