Question

\(\sin 2 \theta = 2 \sin \theta\) is true, when \(\theta\) is equal to

• A. 90\(^\circ\)
• B. 60\(^\circ\)
• C. 45\(^\circ\)
• D. 0\(^\circ\)

Hint:

An easy way is to test Option (D) directly:
LHS = \(\sin (2 \times 0^\circ) = \sin 0^\circ = 0\)
RHS = \(2 \sin 0^\circ = 2(0) = 0\)
Since LHS = RHS, \(0^\circ\) is correct!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We need to find the value of \(\theta\) from the given options for which the trigonometric identity \(\sin 2 \theta = 2 \sin \theta\) holds true.

Step 2: Key Formula or Approach:
Substitute the value of \(\theta\) from each option into the given equation to check which one satisfies it, or use the double-angle formula \(\sin 2\theta = 2\sin\theta\cos\theta\).

Step 3: Detailed Explanation:

• Let us use the double-angle identity:
\[ \sin 2\theta = 2\sin\theta\cos\theta \]

• Substitute this into the given equation:
\[ 2\sin\theta\cos\theta = 2\sin\theta \]

• Rearrange the equation:
\[ 2\sin\theta\cos\theta - 2\sin\theta = 0 \]
\[ 2\sin\theta(\cos\theta - 1) = 0 \]

• This gives two possibilities:
Either \(\sin\theta = 0 \implies \theta = 0^\circ\)
Or \(\cos\theta = 1 \implies \theta = 0^\circ\)

• From the options, only \(\theta = 0^\circ\) is present.


Step 4: Final Answer:
The equation is true when \(\theta = 0^\circ\).