if cos ` x= (-sqrt(15))/(4) " and " (pi)/(2) lt x lt pi` find the value of sin x .

Answer

To find the value of \( \sin x \) given that \( \cos x = -\frac{\sqrt{15}}{4} \) and \( \frac{\pi}{2} < x < \pi \), we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know from the Pythagorean identity that:
\[ \sin^2 x + \cos^2 x = 1 \]
We can rearrange this to find \( \sin^2 x \):
\[ \sin^2 x = 1 - \cos^2 x \]

### Step 2: Substitute the value of \( \cos x \)
Substituting \( \cos x = -\frac{\sqrt{15}}{4} \) into the equation:
\[ \sin^2 x = 1 - \left(-\frac{\sqrt{15}}{4}\right)^2 \]

### Step 3: Calculate \( \cos^2 x \)
Calculating \( \left(-\frac{\sqrt{15}}{4}\right)^2 \):
\[ \cos^2 x = \left(-\frac{\sqrt{15}}{4}\right)^2 = \frac{15}{16} \]

### Step 4: Substitute \( \cos^2 x \) back into the equation
Now substituting back into the equation for \( \sin^2 x \):
\[ \sin^2 x = 1 - \frac{15}{16} \]

### Step 5: Simplify the equation
Calculating \( 1 - \frac{15}{16} \):
\[ \sin^2 x = \frac{16}{16} - \frac{15}{16} = \frac{1}{16} \]

### Step 6: Find \( \sin x \)
Taking the square root of both sides:
\[ \sin x = \pm \frac{1}{4} \]

### Step 7: Determine the correct sign
Since \( x \) is in the second quadrant (where \( \frac{\pi}{2} < x < \pi \)), the sine function is positive in this quadrant. Therefore:
\[ \sin x = \frac{1}{4} \]

### Final Answer
Thus, the value of \( \sin x \) is:
\[ \sin x = \frac{1}{4} \]

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