If `int_(pi//2)^(theta) sin x dx=sin 2 theta` then the of `theta` satisfying `0 lt theta lt pi`, is

Options

A. `3pi//2`
B. `pi//6`
C. `5pi//6`
D. `pi//2`

Explanation

To solve the problem, we need to evaluate the integral and find the values of \(\theta\) that satisfy the equation:

\[ \int_{\frac{\pi}{2}}^{\theta} \sin x \, dx = \sin(2\theta) \]

### Step 1: Evaluate the integral

First, we calculate the integral \(\int \sin x \, dx\):

\[ \int \sin x \, dx = -\cos x + C \]

Now, we can evaluate the definite integral from \(\frac{\pi}{2}\) to \(\theta\):

\[ \int_{\frac{\pi}{2}}^{\theta} \sin x \, dx = \left[-\cos x\right]_{\frac{\pi}{2}}^{\theta} = -\cos(\theta) - (-\cos(\frac{\pi}{2})) \]

Since \(\cos(\frac{\pi}{2}) = 0\), we have:

\[ -\cos(\theta) + 0 = -\cos(\theta) \]

Thus, the integral simplifies to:

\[ -\cos(\theta) = \sin(2\theta) \]

### Step 2: Rewrite the equation

Now we rewrite the equation:

\[ -\cos(\theta) = \sin(2\theta) \]

Using the double angle identity for sine, we know that:

\[ \sin(2\theta) = 2\sin(\theta)\cos(\theta) \]

So we can rewrite our equation as:

\[ -\cos(\theta) = 2\sin(\theta)\cos(\theta) \]

### Step 3: Rearranging the equation

Rearranging gives us:

\[ \cos(\theta) + 2\sin(\theta)\cos(\theta) = 0 \]

Factoring out \(\cos(\theta)\):

\[ \cos(\theta)(1 + 2\sin(\theta)) = 0 \]

### Step 4: Solving for \(\theta\)

This gives us two cases to consider:

1. \(\cos(\theta) = 0\)
2. \(1 + 2\sin(\theta) = 0\)

**Case 1:**

\(\cos(\theta) = 0\)

This occurs when:

\[ \theta = \frac{\pi}{2} \]

**Case 2:**

\(1 + 2\sin(\theta) = 0\)

This simplifies to:

\[ \sin(\theta) = -\frac{1}{2} \]

The sine function is negative in the third and fourth quadrants. The solutions in the range \(0 < \theta < \pi\) are:

\[ \theta = \frac{7\pi}{6} \quad \text{(not valid since it exceeds }\pi\text{)} \]
\[ \theta = \frac{5\pi}{6} \quad \text{(valid)} \]

### Step 5: Conclusion

Thus, the values of \(\theta\) satisfying the original equation in the interval \(0 < \theta < \pi\) are:

\[ \theta = \frac{\pi}{2} \quad \text{and} \quad \theta = \frac{5\pi}{6} \]

### Final Answer

The values of \(\theta\) satisfying the equation are:

\[ \theta = \frac{\pi}{2}, \frac{5\pi}{6} \]