Question

Given that $\sin 2\alpha = \frac{\sqrt{3}}{2}$, the value of $\sin 3\alpha$ is :

• A. $\frac{3\sqrt{3}}{4}$
• B. $\frac{1}{2}$
• C. $1$
• D. $\frac{\sqrt{3}}{4}$

Hint:

Always keep standard trigonometric tables memorized:
$\sin 30^\circ = \frac{1}{2}$, $\sin 45^\circ = \frac{1}{\sqrt{2}}$, $\sin 60^\circ = \frac{\sqrt{3}}{2}$, and $\sin 90^\circ = 1$.
Knowing these standard values allows you to quickly work back and forth between angles and their trigonometric ratios.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question is from the topic "Introduction to Trigonometry".
We are given a trigonometric equation involving an angle $\alpha$: $\sin 2\alpha = \frac{\sqrt{3}}{2}$.
We need to find the value of another trigonometric expression, $\sin 3\alpha$.

Step 2: Key Formula or Approach:
We will find the value of the angle $\alpha$ using standard trigonometric values:

• We know that $\sin(\theta) = \frac{\sqrt{3}}{2}$ has standard acute angle solutions.

• Specifically, the sine of $60^\circ$ is $\frac{\sqrt{3}}{2}$:
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \]

• By equating $2\alpha$ to the standard angle, we can solve for $\alpha$.

• Once $\alpha$ is determined, we can calculate $3\alpha$ and find its sine value.

Step 3: Detailed Explanation:

• Write down the given equation:
\[ \sin 2\alpha = \frac{\sqrt{3}}{2} \]

• Identify the angle whose sine value is $\frac{\sqrt{3}}{2}$ in the range of acute angles:
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \]

• Equate the angles:
\[ 2\alpha = 60^\circ \]

• Divide by 2 to find the value of $\alpha$:
\[ \alpha = 30^\circ \]

• Now, we need to find the value of the expression $\sin 3\alpha$.
Substitute the value of $\alpha = 30^\circ$ into this expression:
\[ \sin 3\alpha = \sin(3 \times 30^\circ) \] \[ \sin 3\alpha = \sin(90^\circ) \]

• We know from standard trigonometric values that:
\[ \sin(90^\circ) = 1 \] Therefore:
\[ \sin 3\alpha = 1 \]

Step 4: Final Answer:
The value of $\sin 3\alpha$ is $1$, which corresponds to Option (C).