Start with the equation \(15 \sin^2 x = 13 + \cos x\).

Use the identity \(\sin^2 x = 1 - \cos^2 x\) to rewrite the equation:

\(15(1 - \cos^2 x) = 13 + \cos x\).

Expand and simplify:

\(15 - 15 \cos^2 x = 13 + \cos x\).

Rearrange to form a quadratic equation:

\(15 \cos^2 x + \cos x - 2 = 0\).

Factor the quadratic equation:

\((5 \cos x + 2)(3 \cos x - 1) = 0\).

Solve for \(\cos x\):

\(5 \cos x + 2 = 0\) gives \(\cos x = -\frac{2}{5}\).

\(3 \cos x - 1 = 0\) gives \(\cos x = \frac{1}{3}\).

Find the angles \(x\) for each value of \(\cos x\) within the range \(0^\circ \leq x \leq 180^\circ\):

For \(\cos x = -\frac{2}{5}\), \(x \approx 113.6^\circ\).

For \(\cos x = \frac{1}{3}\), \(x \approx 70.5^\circ\).