(b) \( 2 \cos 2 \theta+3 \sin \theta-2=0 \)

To solve the equation \( 2 \cos 2 \theta + 3 \sin \theta - 2 = 0 \), we can use the double angle formula for cosine, which states that \( \cos 2\theta = 1 - 2\sin^2\theta \). This can help us express everything in terms of sine. Substituting, we get \( 2(1 - 2\sin^2\theta) + 3\sin\theta - 2 = 0 \), which simplifies to \( -4\sin^2\theta + 3\sin\theta = 0 \). Factor out \(\sin\theta\) to get \(\sin\theta(3 - 4\sin\theta) = 0\). Thus, \(\sin\theta = 0\) or \(\sin\theta = \frac{3}{4}\). The \( \sin\theta = 0 \) gives us solutions at integer multiples of \(\pi\) (i.e., \(\theta = n\pi\) for \(n \in \mathbb{Z}\)). For \( \sin\theta = \frac{3}{4} \), this would yield additional solutions involving the arcsine function, like \( \theta = \arcsin\left(\frac{3}{4}\right) + 2k\pi \) and \( \theta = \pi - \arcsin\left(\frac{3}{4}\right) + 2k\pi\) for integers \(k\). Breaking this into more palatable parts for us math enthusiasts, it's amazing that the sine function can yield so many angles that share the same property! Finding the exact decimal value of \(\arcsin\left(\frac{3}{4}\right)\) might require a calculator, but it’s a great excuse to whip out those tech tools! While many people jump straight into solving trigonometric equations using trial-and-error, it’s all about understanding the properties of functions and how they relate. The common mistake often lies in neglecting to explore all possible angles, like those in different quadrants or relating back to the periodic nature of trigonometric functions. Taking that extra moment to consider the unit circle can save you from overlooking some delightful solutions!