`if A={theta: cos theta gt -(1)/(2),0 le theta le pi }and `
`B={theta: sin thetagt(1)/(2) ,(pi)/(3) lt theta le pi}`then

Options

A. `A cap B= {theta:pi//3le theta le 2pi//3}`
B. `A cap B= {theta:-pi//3le theta le 2pi//3}`
C. `A cap B= {theta:-5pi//6le theta le 5pi//6}`
D. `A cup B= {theta:0 letheta le pi//6}`

Answer

To solve the problem, we need to find the intersection of the sets A and B defined by the given conditions.

### Step 1: Define Set A
Set A is defined as:
\[ A = \{ \theta : \cos \theta > -\frac{1}{2}, 0 \leq \theta \leq \pi \} \]

To find the values of \(\theta\) that satisfy \(\cos \theta > -\frac{1}{2}\), we need to determine the angles where the cosine function is greater than \(-\frac{1}{2}\) within the interval \([0, \pi]\).

### Step 2: Find the critical points for Set A
The cosine function equals \(-\frac{1}{2}\) at:
\[ \theta = \frac{2\pi}{3} \]

Thus, \(\cos \theta > -\frac{1}{2}\) when:
\[ 0 \leq \theta < \frac{2\pi}{3} \]

So, we can express Set A as:
\[ A = \{ \theta : 0 \leq \theta < \frac{2\pi}{3} \} \]

### Step 3: Define Set B
Set B is defined as:
\[ B = \{ \theta : \sin \theta > \frac{1}{2}, \frac{\pi}{3} < \theta \leq \pi \} \]

To find the values of \(\theta\) that satisfy \(\sin \theta > \frac{1}{2}\), we need to determine the angles where the sine function is greater than \(\frac{1}{2}\).

### Step 4: Find the critical points for Set B
The sine function equals \(\frac{1}{2}\) at:
\[ \theta = \frac{\pi}{6} \]

Thus, \(\sin \theta > \frac{1}{2}\) when:
\[ \frac{\pi}{6} < \theta < \frac{5\pi}{6} \]

However, since we are restricted to \(\frac{\pi}{3} < \theta \leq \pi\), we need to find the overlap:
- The interval for \(\sin \theta > \frac{1}{2}\) is \((\frac{\pi}{6}, \frac{5\pi}{6})\).
- The interval for Set B is \((\frac{\pi}{3}, \pi]\).

### Step 5: Determine the intersection of Set A and Set B
Now we need to find the intersection of Set A and Set B:
- Set A: \(0 \leq \theta < \frac{2\pi}{3}\)
- Set B: \(\frac{\pi}{3} < \theta \leq \pi\)

The intersection will be:
\[ A \cap B = \{ \theta : \frac{\pi}{3} < \theta < \frac{2\pi}{3} \} \]

### Final Answer
Thus, the intersection of sets A and B is:
\[ A \cap B = \left( \frac{\pi}{3}, \frac{2\pi}{3} \right) \]

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