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 Given Conditions:
Given:
tan(A + 2B) = √3
sin(2A + B) = 1/√2

Also, A, B, A + 2B and 2A + B are acute angles.
So we take only acute angle values of standard trigonometric ratios.

Step 2: Using Standard Values:
We know:
tan 60° = √3
sin 45° = 1/√2

Since angles are acute,
A + 2B = 60° …(1)
2A + B = 45° …(2)

Step 3: Solving the System of Equations:
From (1):
A + 2B = 60

From (2):
2A + B = 45

Multiply (1) by 2:
2A + 4B = 120 …(3)

Now subtract (2) from (3):
(2A + 4B) − (2A + B) = 120 − 45
3B = 75
B = 25°

Substitute B = 25° in (1):
A + 2(25) = 60
A + 50 = 60
A = 10°

Step 4: Verification:
A + 2B = 10 + 50 = 60° ✔
2A + B = 20 + 25 = 45° ✔

Final Answer:
A = 10°
B = 25°