\( \therefore x+30^{\circ}=60^{\circ}+k \cdot 360^{\circ} \) or \( x+30^{\circ}=120^{\circ}+k \cdot 360^{\circ} ; k \in \mathbb{Z} \) \( \therefore x=30^{\circ}+k \cdot 360^{\circ} \) or \( x=90^{\circ}+k \cdot 360^{\circ} ; k \in \mathbb{Z} \) EXERCISE 10 Solve for \( x \) : \( \begin{array}{ll}\text { (a) } 3 \sin 2 x-2 \sin x=0 & \text { (b) } \cos 2 x+5 \cos x-2=0 ; x \in\left[0^{\circ} ; 360\right.\end{array} \)

To tackle the equation \( 3 \sin 2x - 2 \sin x = 0 \), we can utilize the double angle identity for sine, which states that \( \sin 2x = 2 \sin x \cos x \). Substituting this in gives \( 3(2 \sin x \cos x) - 2 \sin x = 0 \). Factoring out \( \sin x \) leads us to \( \sin x (6 \cos x - 2) = 0 \). Thus, \( \sin x = 0 \) or \( 6 \cos x - 2 = 0 \). Solving these equations reveals \( x = 0^{\circ}, 180^{\circ} \) or \( \cos x = \frac{1}{3} \), yielding \( x = 70.53^{\circ} \) or \( x = 289.47^{\circ} \). For \( (b) \), the equation is \( \cos 2x + 5 \cos x - 2 = 0 \). Again employing the double angle identity, we substitute to get \( 2 \cos^2 x - 1 + 5 \cos x - 2 = 0 \), simplifying to \( 2 \cos^2 x + 5 \cos x - 3 = 0 \). By using the quadratic formula \( \cos x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), we find possible values for \( \cos x \), which can then be converted back to \( x \) using the range constraints. Make sure to check for angles within \( [0^{\circ}, 360^{\circ}) \). Finding the roots of trigonometric equations is a marvelous journey! Did you know that trigonometric functions are periodic, so understanding this concept can help you predict certain values even before solving? This periodicity hints that multiple solutions can appear across different intervals simply by adding multiples of \( 360^{\circ} \). When solving equations, common pitfalls include forgetting the constraints of the angle ranges or overlooking solutions that come from identities or factorizations. It can be easy to miss \( \sin x = 0 \) or how many solutions can emerge from a quadratic approach. Remember: always double-check your work!