To solve: The equation 4 \cos \theta \sin \theta + 3 \cos \theta = 0

Approach:
The domain of the trigonometry function of ${\displaystyle \cos \theta }$ is lies between ${\displaystyle [-1,1]}$. No solution exists beyond this domain.
Obtain factor of given equation and simply the factor equations to obtain solutions.
Cosine and sine have period ${\displaystyle 2\pi }$, thus find the solution in any interval of length ${\displaystyle 2\pi }$.
Sine function is positive in first and second quadrant. Cosine function is positive in first and fourth quadrant.
Calculation:
Consider the trigonometry equation.
${\displaystyle 4\cos \theta \sin \theta +3\cos \theta =0}$
The factors of above equation are obtained by,
${\displaystyle 4\cos \theta \sin \theta +3\cos \theta =0}$
${\displaystyle \cos \theta (4\sin \theta +3)=0}$
The factors are,
${\displaystyle 4\sin \theta +3=0\dots \left(1\right)}$
${\displaystyle \cos \theta =0\dots \left(2\right)}$
The solution obtained for the factor in which sine and cosine functions are involved so we will get the solution in the interval of ${\displaystyle [0,2\pi ]}$
Substract 3 both sides in equation (1).
${\displaystyle 4\sin \theta =-3}$
Divide by 4 both sides in equation (1).
${\displaystyle \sin \theta =-\frac{3}{4}}$
Multiple by ${\displaystyle {\sin }^{-1}}$ both sides in equation (1).
${\displaystyle {\sin }^{-1}\sin \theta ={\sin }^{-1}(-\frac{3}{4})}$
${\displaystyle \theta ={\sin }^{-1}(-\frac{3}{4})}$
${\displaystyle =4,5.435}$
Here solutions are in radian.
The solution repeats value of the equation at every length of ${\displaystyle \pi }$ in the interval ${\displaystyle [0,2\pi ]}$.
We will get all solutions of the equation by adding integer multiples of ${\displaystyle \pi }$ to these solutions:
${\displaystyle \theta =4+2k\pi }$
${\displaystyle \theta =5.435+2k\pi }$
Consider the factor in equation (2).
${\displaystyle \cos \theta =0}$
${\displaystyle \theta =\frac{\pi }{2},\frac{3\pi }{2}}$
The solution repeats value of the equation at every length of ${\displaystyle \pi }$ in the interval ${\displaystyle [0,2\pi ]}$
We will get all solutions of the equation by adding integer multiples of ${\displaystyle \pi }$ to these solutions: ${\displaystyle \theta =\frac{\pi }{2}+k\pi }$
Therefore, the solutions of the trigonometry equation ${\displaystyle 4\cos \theta \sin \theta +3\cos \theta =0}$ are ${\displaystyle \theta =\frac{\pi }{2}+k\pi ,\theta =4+2k\pi }$ and ${\displaystyle \theta =5.435+2k\pi }$
Conclusion:
Hence, the solutions of the trigonometry equation ${\displaystyle 4\cos \theta \sin \theta +3\cos \theta =0}$ are ${\displaystyle \theta =\frac{\pi }{2}+k\pi ,\theta =4+2k\pi }$ and ${\displaystyle \theta =5.435+2k\pi }$