Find the value of `3sin20^(@)-4sin^(3)20^(@).`

Options

A. `(sqrt(3))/(2)`
B. `(1)/(2)`
C. `(1)/(sqrt(2))`
D. `(2)/(sqrt(3))`

Answer

To find the value of \( 3\sin(20^\circ) - 4\sin^3(20^\circ) \), we can use the identity for \( \sin(3a) \):

### Step-by-step Solution:

1. **Identify the Identity**:
We know that:
\[ \sin(3a) = 3\sin(a) - 4\sin^3(a) \]
Here, we can set \( a = 20^\circ \).

2. **Substitute \( a \) into the Identity**:
Using the identity, we can rewrite our expression:
\[ \sin(3 \times 20^\circ) = 3\sin(20^\circ) - 4\sin^3(20^\circ) \]
This simplifies to:
\[ \sin(60^\circ) = 3\sin(20^\circ) - 4\sin^3(20^\circ) \]

3. **Calculate \( \sin(60^\circ) \)**:
We know that:
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \]

4. **Set the Equation**:
Now we have:
\[ \frac{\sqrt{3}}{2} = 3\sin(20^\circ) - 4\sin^3(20^\circ) \]

5. **Conclusion**:
Therefore, the value of \( 3\sin(20^\circ) - 4\sin^3(20^\circ) \) is equal to \( \sin(60^\circ) \), which is:
\[ \frac{\sqrt{3}}{2} \]

### Final Answer:
\[ 3\sin(20^\circ) - 4\sin^3(20^\circ) = \frac{\sqrt{3}}{2} \]