Evaluate `sin(1/2cot^(-1)((-3)/(4))`

Answer

To evaluate \( \sin\left(\frac{1}{2} \cot^{-1}\left(-\frac{3}{4}\right)\right) \), we can follow these steps:

### Step 1: Define the angle
Let \( \theta = \cot^{-1}\left(-\frac{3}{4}\right) \). This means that \( \cot(\theta) = -\frac{3}{4} \).

### Step 2: Convert to sine and cosine
Using the definition of cotangent, we have:
\[ \cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} = -\frac{3}{4} \]
This implies that we can represent this in a right triangle where the adjacent side is -3 and the opposite side is 4. To find the hypotenuse, we use the Pythagorean theorem:
\[ \text{hypotenuse} = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Find sine and cosine
Now we can find \( \sin(\theta) \) and \( \cos(\theta) \):
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{5} \]
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{-3}{5} \]

### Step 4: Use the half-angle formula
Next, we need to find \( \sin\left(\frac{1}{2} \theta\right) \). We can use the half-angle formula:
\[ \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}} \]
Substituting \( \cos(\theta) = -\frac{3}{5} \):
\[ \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \left(-\frac{3}{5}\right)}{2}} = \sqrt{\frac{1 + \frac{3}{5}}{2}} = \sqrt{\frac{\frac{5}{5} + \frac{3}{5}}{2}} = \sqrt{\frac{\frac{8}{5}}{2}} = \sqrt{\frac{8}{10}} = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}} \]

### Step 5: Final result
Thus, the value of \( \sin\left(\frac{1}{2} \cot^{-1}\left(-\frac{3}{4}\right)\right) \) is:
\[ \frac{2}{\sqrt{5}} \]