If `sinx=1/(3)`, then the value of `sin3x` is

Options

A. 1
B. 0
C. `23/(27)`
D. `-23/(27)`

Answer

To find the value of \( \sin 3x \) given that \( \sin x = \frac{1}{3} \), we can use the formula for \( \sin 3\theta \):

\[ \sin 3\theta = 3\sin \theta - 4\sin^3 \theta \]

### Step 1: Substitute \( \sin x \) into the formula
Let \( \theta = x \). Then we have:

\[ \sin 3x = 3\sin x - 4\sin^3 x \]

Substituting \( \sin x = \frac{1}{3} \):

\[ \sin 3x = 3\left(\frac{1}{3}\right) - 4\left(\frac{1}{3}\right)^3 \]

### Step 2: Simplify the expression
Calculating \( 3\left(\frac{1}{3}\right) \):

\[ 3\left(\frac{1}{3}\right) = 1 \]

Now, calculate \( \left(\frac{1}{3}\right)^3 \):

\[ \left(\frac{1}{3}\right)^3 = \frac{1}{27} \]

Now substitute this back into the equation:

\[ \sin 3x = 1 - 4\left(\frac{1}{27}\right) \]

### Step 3: Calculate \( 4\left(\frac{1}{27}\right) \)
Calculating \( 4\left(\frac{1}{27}\right) \):

\[ 4\left(\frac{1}{27}\right) = \frac{4}{27} \]

### Step 4: Final calculation
Now substitute this back into the equation for \( \sin 3x \):

\[ \sin 3x = 1 - \frac{4}{27} \]

To perform the subtraction, we need a common denominator:

\[ 1 = \frac{27}{27} \]

Thus,

\[ \sin 3x = \frac{27}{27} - \frac{4}{27} = \frac{27 - 4}{27} = \frac{23}{27} \]

### Final Answer
The value of \( \sin 3x \) is:

\[ \sin 3x = \frac{23}{27} \]

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