Calculate, in degrees, the principal values of $x$ which satisfy
$2 \sin 2x + 3 \cos x = 0$ .

1. Apply the double angle identity for sine

$2(2 \sin x \cos x) + 3 \cos x = 0$
$4 \sin x \cos x + 3 \cos x = 0$

2. Factorise the equation to find the possible trigonometric values

$\cos x (4 \sin x + 3) = 0$
$\cos x = 0 \quad \text{or} \quad \sin x = -\frac{3}{4}$

3. Calculate the principal values for each case

For $\cos x = 0$, the principal value is:
$x = \cos^{-1}(0) = 90^\circ$

For $\sin x = -0.75$, the principal value is:
$x = \sin^{-1}(-0.75) = -48.590^\circ \approx -48.6^\circ \text{ (to 1 d.p.)}$