Given question is : $$\sin(5\alpha + \theta) = \cos(\theta – 3\alpha)$$
We are to find the least positive value of $\alpha$ for which above equation holds.
The way I did is as, $$\sin5\alpha \cos\theta + \cos5\alpha \sin\theta = \cos\theta \cos3\alpha + \sin\theta \sin3\alpha$$
Now for this to be true $$\sin5\alpha = \cos3\alpha$$ and $$\cos5\alpha = \sin3\alpha$$
How do I find the value of $\alpha$ that satisfies the above criteria?
Best Answer
Divide the two relations, after noting that none among $\sin5\alpha$, $\cos5\alpha$, $\sin3\alpha$ and $\cos3\alpha$ can be zero, if the relations you have are to hold. Then $$ \tan5\alpha=\cot3\alpha=\tan\left(\frac{\pi}{2}-3\alpha\right) $$ Recall that $\tan x=\tan y$ if and only if $x=y+k\pi$, for some integer $k$.