Question

For acute angles A and B and \(A + 2B\) and \(2A + B\) are acute if \(\tan (A + 2B) = \sqrt{3}\) and \(\sin (2A + B) = \frac{1}{\sqrt{2}}\), then find the measures of angles A and B.}

Hint:

Always check if the calculated angles satisfy the "acute" condition mentioned in the question. Here, \(A+2B = 10+50 = 60^{\circ}\) and \(2A+B = 20+25 = 45^{\circ}\), both are acute.

Answer 1

Step 1: Understanding the Concept:
We use the inverse values of trigonometric functions for standard angles to form a system of two linear equations in terms of \(A\) and \(B\).
Step 2: Detailed Explanation:
From \(\tan (A + 2B) = \sqrt{3}\):
Since \(\tan 60^{\circ} = \sqrt{3}\), we have:
\[ A + 2B = 60^{\circ} \quad \dots(1) \]
From \(\sin (2A + B) = \frac{1}{\sqrt{2}}\):
Since \(\sin 45^{\circ} = \frac{1}{\sqrt{2}}\), we have:
\[ 2A + B = 45^{\circ} \quad \dots(2) \]
Solving the equations:
Multiply equation (2) by 2:
\[ 4A + 2B = 90^{\circ} \quad \dots(3) \]
Subtract equation (1) from (3):
\[ (4A + 2B) - (A + 2B) = 90^{\circ} - 60^{\circ} \]
\[ 3A = 30^{\circ} \implies A = 10^{\circ} \]
Substitute \(A = 10^{\circ}\) in equation (2):
\[ 2(10^{\circ}) + B = 45^{\circ} \]
\[ 20^{\circ} + B = 45^{\circ} \implies B = 25^{\circ} \]
Step 3: Final Answer:
The measures of the angles are \(A = 10^{\circ}\) and \(B = 25^{\circ}\).