Question

How can we obtain the general expression of all the angles having a same sine ?

Answer 1

Obtain ANY one angle \(\alpha\) whose sine we have to consider. We have to find the general solution for \(\theta\) in the following equation:\[\sin \theta = \sin \alpha \]\[\Rightarrow \sin \theta - \sin \alpha = 0\]\[\Rightarrow 2 \cos \left(\frac{\theta + \alpha}2\right)\sin \left(\frac{\theta - \alpha }2\right)=0\]This means\[\theta - \alpha = 2n_1\pi\]or\[\theta + \alpha = (2n_2+1)\pi \]These two cases can be written in a condensed form as follows:\[\boxed{\theta = n\pi + (-1)^n \alpha}\tag{general solution}\]

Answer 2

can we derive it through any geometrical method ??

Answer 3

Yes, definitely. Look at the unit circle.

Answer 4

Exactly.