A={x:cos⁡x>−1/2, 0≤x≤π},B={x:sin⁡x>1/2,π/3≤x≤π} then

A={x:cos⁡x>−1/2,0≤x≤π}={x:0≤x≤2π/3}B=x:sin⁡x>12,π3≤x≤π=x:π3≤x≤5π6∴A∩B=π3,2π3

A={x:cos⁡x>−1/2, 0≤x≤π},B={x:sin⁡x>1/2,π/3≤x≤π} then

A={x:cos⁡x>−1/2,0≤x≤π}={x:0≤x≤2π/3}B=x:sin⁡x>12,π3≤x≤π=x:π3≤x≤5π6∴A∩B=π3,2π3

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$A = {x : \cos x > - 1 / 2, 0 \leq x \leq π}, B = {x : \sin x > 1 / 2, π / 3 \leq x \leq π}$ then

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$A \cap B = [\frac{π}{3}, \frac{2 π}{3}]$

$A \cap B = [- \frac{π}{3}, \frac{2 π}{3}]$

$A \cup B = [- \frac{5 π}{6}, \frac{5 π}{6}]$

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Detailed Solution

$\begin{array}{l} A = {x : \cos x > - 1 / 2, 0 \leq x \leq π} = {x : 0 \leq x \leq 2 π / 3} \\ B = \{x : \sin x > \frac{1}{2}, \frac{π}{3} \leq x \leq π\} = \{x : \frac{π}{3} \leq x \leq \frac{5 π}{6}\} \\ ∴ A \cap B = [\frac{π}{3}, \frac{2 π}{3}] \end{array}$