Question

Choose the correct option and justify your choice: $\sin 2{\text{A}} = 2\sin {\text{A}}$ is true when ${\text{A}}$ is:
(A) $0^\circ $
(B) $30^\circ $
(C) $45^\circ $
(D) $60^\circ $

Answer 1

Hint: In this question we have to use the formula of \[\sin 2{\text{A}}\] and then we will put this formula in the equation which is given in the question. Now, we will simplify the equation. We also need to remember that the value of $\cos $ is $1$ when the angle is $0^\circ $.

Complete step-by-step solution:
The given equation is $\sin 2{\text{A}} = 2\sin {\text{A}}$ . Now, put the formula \[\sin 2{\text{A}} = 2\sin {\text{A cosA}}\] in the equation given in the question and try to simplify it.
Therefore, we can write $\sin 2{\text{A}} = 2\sin {\text{A}}$ as follows:
$
  \sin 2{\text{A}} = 2\sin {\text{A}} \\
   \Rightarrow 2\sin {\text{A cosA}} = 2\sin {\text{A ……………..(1)}}
 $
Now, just cancel out $2\sin {\text{A}}$from both the sides of the equation $(1).$ Therefore, we will get
$ \Rightarrow \cos {\text{A}} = 1$
Now, we know that the value of $\cos $ is $1$ when the angle ${\text{A}}$ is $0^\circ $ i.e. $\cos 0 = 1$
Therefore, we get $\sin 2{\text{A}} = 2\sin {\text{A}}$ when the angle the value of the angle ${\text{A}}$ is $0^\circ $ .

Hence, the correct answer is option$\left( {\text{A}} \right)$.

Note: We can also solve the question putting the value of angle ${\text{A}}$ available in the options.
If we put the value of angle ${\text{A}} = 0^\circ $ in the equation $\sin 2{\text{A}} = 2\sin {\text{A}}$ . Then we will get
$
  \sin 2\left( {0^\circ } \right) = 2\sin \left( {0^\circ } \right) \\
   \Rightarrow \sin 0^\circ = 2\sin 0^\circ \\
   \Rightarrow 0 = 0
 $
Therefore, we are getting the same answer by using different methods.
Hence, the correct answer is option $\left( {\text{A}} \right)$.