The value of `3sin 30^(@)-4sin^(3)30^(@)` is :

Options

A. `sin60^(@)`
B. `sin90^(@)`
C. 0
D. None of these

Answer

To solve the expression \( 3 \sin 30^\circ - 4 \sin^3 30^\circ \), we will follow these steps:

### Step 1: Find the value of \( \sin 30^\circ \)
We know that:
\[ \sin 30^\circ = \frac{1}{2} \]

### Step 2: Substitute \( \sin 30^\circ \) into the expression
Now, we substitute \( \sin 30^\circ \) into the expression:
\[ 3 \sin 30^\circ - 4 \sin^3 30^\circ = 3 \left(\frac{1}{2}\right) - 4 \left(\frac{1}{2}\right)^3 \]

### Step 3: Calculate \( \sin^3 30^\circ \)
Next, we calculate \( \sin^3 30^\circ \):
\[ \sin^3 30^\circ = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]

### Step 4: Substitute \( \sin^3 30^\circ \) back into the expression
Now we substitute this value back into the expression:
\[ 3 \left(\frac{1}{2}\right) - 4 \left(\frac{1}{8}\right) \]

### Step 5: Simplify the expression
Calculating each term:
\[ 3 \left(\frac{1}{2}\right) = \frac{3}{2} \]
\[ 4 \left(\frac{1}{8}\right) = \frac{4}{8} = \frac{1}{2} \]

Now substituting these values:
\[ \frac{3}{2} - \frac{1}{2} \]

### Step 6: Find a common denominator and subtract
To subtract these fractions, we can write:
\[ \frac{3}{2} - \frac{1}{2} = \frac{3 - 1}{2} = \frac{2}{2} = 1 \]

### Final Answer
Thus, the value of \( 3 \sin 30^\circ - 4 \sin^3 30^\circ \) is:
\[ \boxed{1} \]