Question : If $\sin (A-B)=\frac{1}{2}$ and $\sin (A+B)=1$, where A and B are positive acute angles and $ A \geq B$ then A and B are respectively:
Option 1: 80° and 10°
Option 2: 60° and 30°
Option 3: 70° and 20°
Option 4: 50° and 40°
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Correct Answer: 60° and 30°
Solution :
$\sin (A-B)=\frac{1}{2}$
⇒ $\sin (A-B)=\sin 30^\circ$
⇒ $(A - B) = 30^\circ$
⇒ $A = 30^\circ + B$
And
$\sin (A+B)=1$
⇒ $\sin (A+B)=\sin 90^\circ$
⇒ $(A+B) = 90^\circ$
⇒ $30^\circ + B + B = 90^\circ$
⇒ $2B = 60^\circ$
⇒ $B = 30^\circ$
$\therefore A = 30^\circ + 30^\circ = 60^\circ$
Hence, the correct answer is 60° and 30°.
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