Question : If $\cos (A-B)=\frac{\sqrt{3}}{2}$, and $\cos (A+B)=0$, where $A$ and $B$ are positive acute angles and $A \geq B$, then the measures of $A$ and $B$ are:
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 :
Given, $\cos (A-B)=\frac{\sqrt{3}}{2}$ and $\cos (A+B)=0$
Since $A$ and $B$ are acute positive angles,
$\cos (A-B)=\frac{\sqrt{3}}{2}$
⇒ $(A-B) = \cos^{-1} \frac{\sqrt{3}}{2}$
⇒ $(A-B) = 30°$ ------------------ (1)
$\cos (A+B)=0$
⇒ $(A+B) = \cos^{-1} 0$
⇒ $(A+B) = 90°$ --------------------- (2)
Solving (1) and (2), we get,
$A = 60°$ and $B = 30°$
Hence, the correct answer is 60° and 30°.
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