Question

If $\cos A = \dfrac{\sqrt{3}}{2}$, then the value of $\sin 2A$ will be:

• A. 1
• B. 0
• C. $\dfrac{1}{2}$
• D. $\dfrac{\sqrt{3}}{2}$

Hint:

Use $\sin 2A = 2 \sin A \cos A$ and the Pythagoras identity $\sin^2 A + \cos^2 A = 1$ to relate both functions.

Answer 1

The Correct Option is A

Step 1: Recall trigonometric identity.
\[ \sin 2A = 2 \sin A \cos A \]
Step 2: Find $\sin A$.
Given $\cos A = \dfrac{\sqrt{3}}{2}$, therefore $\sin A = \dfrac{1}{2}$ (since $\sin^2 A + \cos^2 A = 1$).

Step 3: Substitute values.
\[ \sin 2A = 2 \times \dfrac{1}{2} \times \dfrac{\sqrt{3}}{2} = \dfrac{\sqrt{3}}{2} \] Wait, for $\cos A = \dfrac{\sqrt{3}}{2}$, angle $A = 30^\circ$, so $2A = 60^\circ$ and $\sin 60^\circ = \dfrac{\sqrt{3}}{2}$. Hence correct option is (D).

Step 4: Correcting conclusion.
Therefore, the correct value is $\sin 2A = \dfrac{\sqrt{3}}{2}$.