Question

If the trigonometric equation $\sin 2A = 2\sin A$ is true then find the value of $A$.
(A) ${0^ \circ }$
(B) ${30^ \circ }$
(C) ${45^ \circ }$
(D) ${60^ \circ }$

Answer 1

Hint: We know that from a trigonometric we have $\sin 2\theta = 2\sin \theta \cos \theta $. Use this formula in the equation given in the question. Simplify it further and solve it to find the value of angle $A$.

Complete step-by-step solution:
According to the question, we have been given a trigonometric equation and we have to find the value of an unknown angle.
The given trigonometric equation is:
$ \Rightarrow \sin 2A = 2\sin A$.
For solving this equation we’ll apply the trigonometric formula $\sin 2\theta = 2\sin \theta \cos \theta $ on the left hand side of the equation. Doing this, we’ll get:
$ \Rightarrow 2\sin A\cos A = 2\sin A$
We can cancel out 2 from both sides.
$ \Rightarrow \sin A\cos A = \sin A$
Transferring all the terms on one side of the equation, we’ll get:
\[ \Rightarrow \sin A\cos A - \sin A = 0\]
Now simplifying it further by factoring, we’ll get:
\[ \Rightarrow \sin A\left( {\cos A - 1} \right) = 0\]
For this equation to be true, both of its factors can be zero:
\[
   \Rightarrow \sin A = 0{\text{ or }}\cos A - 1 = 0 \\
   \Rightarrow \sin A = 0{\text{ or }}\cos A = 1 \\
   \Rightarrow A = {0^ \circ }{\text{ or }}A = {0^ \circ }
 \]
Thus the value of angle A is ${0^ \circ }$ in both the cases.

(A) is the correct option.

Note: In the above problem, we have used a trigonometric formula for double angle which is:
$ \Rightarrow \sin 2\theta = 2\sin \theta \cos \theta $
Another form of the formula of $\sin 2\theta $ is:
$ \Rightarrow \sin 2\theta = \dfrac{{2\tan \theta }}{{1 + {{\tan }^2}\theta }}$
Some of the other widely used double angle trigonometric formulas are:
$
   \Rightarrow \cos 2\theta = 2{\cos ^2}\theta - 1 \\
   \Rightarrow \cos 2\theta = 1 - 2{\sin ^2}\theta \\
   \Rightarrow \cos 2\theta = {\cos ^2}\theta - {\sin ^2}\theta
 $
So we have the formula of $\cos 2\theta $ in three different forms. We can use any of them as per the requirement of the question.
Similarly the formula for $\tan 2\theta $ is:
$ \Rightarrow \tan 2\theta = \dfrac{{2\tan \theta }}{{1 - {{\tan }^2}\theta }}$