Question

If \(\sin(A + 2B) = 2 \cos 60^\circ\) and \(A = 3B\), find the measures of A and B.

Hint:

When an equation involves \(\sin(\dots) = 1\), immediately replace 1 with \(\sin 90^\circ\) to "remove" the sine function and solve for the angles.

Answer 1

Step 1: Solving OR (B):
1. We know \(2 \cos 60^\circ = 2 \times \frac{1}{2} = 1\).
2. \(\sin(A + 2B) = 1\). Since \(\sin 90^\circ = 1\), then \(A + 2B = 90^\circ\).
3. Substitute \(A = 3B\):
\[ 3B + 2B = 90^\circ \implies 5B = 90^\circ \implies B = 18^\circ \] 4. Find \(A\): \(A = 3 \times 18^\circ = 54^\circ\).
Step 2: Final Answer (OR):
\(A = 54^\circ\) and \(B = 18^\circ\).