1) \( \operatorname{sen} 2 x+\operatorname{sen} x=0 \)

To solve \( \operatorname{sen} 2x + \operatorname{sen} x = 0 \), we can use the double angle identity \( \operatorname{sen} 2x = 2 \operatorname{sen} x \operatorname{cos} x \). This transforms the equation into \( 2 \operatorname{sen} x \operatorname{cos} x + \operatorname{sen} x = 0 \), which can be factored to \( \operatorname{sen} x (2 \operatorname{cos} x + 1) = 0 \). This gives us two cases: \( \operatorname{sen} x = 0 \) and \( 2 \operatorname{cos} x + 1 = 0 \). For the first case, \( \operatorname{sen} x = 0 \) leads to \( x = n\pi \), where \( n \) is any integer. In the second case, solving \( \operatorname{cos} x = -\frac{1}{2} \) gives \( x = \frac{2\pi}{3} + 2k\pi \) or \( x = \frac{4\pi}{3} + 2k\pi \) for any integer \( k \). Hence, the complete solution set is \( x = n\pi \) and \( x = \frac{2\pi}{3} + 2k\pi \) or \( x = \frac{4\pi}{3} + 2k\pi \). For real-world applications, understanding trigonometric equations is essential in fields like physics, engineering, and computer graphics. For instance, they help model periodic phenomena such as sound waves, light waves, or even the motion of pendulums. Being able to solve these equations allows professionals to predict behaviors and design systems effectively. A common mistake in solving trigonometric equations is neglecting the multiple solutions that come from periodic properties. For example, while many might find one or two solutions, the periodic nature of \( \sin(x) \) means each solution repeats every \( 2\pi \) radians. Always remember to account for the general solution, adding multiples of the period to capture all possible angles that satisfy the equation.