`sintheta=(1)/(sqrt(2))`,then find the value of `3sin ^(2)theta-4sin^(3)theta*costheta`.

Answer

To solve the problem where \( \sin \theta = \frac{1}{\sqrt{2}} \) and we need to find the value of \( 3\sin^2\theta - 4\sin^3\theta \cos\theta \), we can follow these steps:

### Step 1: Find \( \sin^2 \theta \)
Given \( \sin \theta = \frac{1}{\sqrt{2}} \), we can find \( \sin^2 \theta \):
\[ \sin^2 \theta = \left( \frac{1}{\sqrt{2}} \right)^2 = \frac{1}{2} \]

### Step 2: Find \( \sin^3 \theta \)
Next, we calculate \( \sin^3 \theta \):
\[ \sin^3 \theta = \left( \frac{1}{\sqrt{2}} \right)^3 = \frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4} \]

### Step 3: Find \( \cos \theta \)
To find \( \cos \theta \), we can use the Pythagorean identity:
\[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{1}{2} = \frac{1}{2} \]
Thus, taking the square root:
\[ \cos \theta = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]

### Step 4: Substitute values into the expression
Now we substitute \( \sin^2 \theta \), \( \sin^3 \theta \), and \( \cos \theta \) into the expression \( 3\sin^2 \theta - 4\sin^3 \theta \cos \theta \):
\[ 3\sin^2 \theta = 3 \times \frac{1}{2} = \frac{3}{2} \]
\[ 4\sin^3 \theta \cos \theta = 4 \times \frac{\sqrt{2}}{4} \times \frac{1}{\sqrt{2}} = 1 \]

### Step 5: Calculate the final result
Now, we can combine these results:
\[ 3\sin^2 \theta - 4\sin^3 \theta \cos \theta = \frac{3}{2} - 1 = \frac{3}{2} - \frac{2}{2} = \frac{1}{2} \]

Thus, the final answer is:
\[ \boxed{\frac{1}{2}} \]