One of the angles of the triangle is

Calculation:

We are given the equation for the ratio of sides using the Sine Rule:

\( \frac{a}{\sin(3x)} = \frac{b}{\sin(x)} \)

\( \frac{a}{b} = \frac{\sin(3x)}{\sin(x)} \)

Step 3: Use the identity for sin(3x), which is \(\sin(3x) = 3\sin(x) - 4\sin^3(x) \), and substitute it into the equation:

\( 2 = \frac{3\sin(x) - 4\sin^3(x)}{\sin(x)} \)

\( 2\sin(x) = 3\sin(x) - 4\sin^3(x) \)

\( -\sin(x) + 4\sin^3(x) = 0 \)

\( \sin(x)(4\sin^2(x) - 1) = 0 \)

We have two possible solutions for this equation:

\( \sin(x) = 0 \), which gives x = \(0^\circ \) (not valid in this case).

\( 4\sin^2(x) - 1 = 0 \), which simplifies to:

\( \sin^2(x) = \frac{1}{4} \quad \Rightarrow \quad \sin(x) = \frac{1}{2} \)

\( x = 30^\circ \)

∴ The correct answer is Option (2)