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

To solve:
The equation $$\displaystyle{4}{\cos{\theta}}{\sin{\theta}}+{3}{\cos{\theta}}={0}$$

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Willie
Approach:
The domain of the trigonometry function of $$\displaystyle{\cos{\theta}}$$ is lies between $$\displaystyle{\left[-{1},{1}\right]}$$. 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,
$$\displaystyle{4}{\cos{\theta}}{\sin{\theta}}+{3}{\cos{\theta}}={0}$$
$$\displaystyle{\cos{\theta}}{\left({4}{\sin{\theta}}+{3}\right)}={0}$$
The factors are,
$$\displaystyle{4}{\sin{\theta}}+{3}={0}\ldots{\left({1}\right)}$$
$$\displaystyle{\cos{\theta}}={0}\ldots{\left({2}\right)}$$
The solution obtained for the factor in which sine and cosine functions are involved so we will get the solution in the interval of $$\displaystyle{\left[{0},{2}\pi\right]}$$
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).
$$\displaystyle{{\sin}^{{-{1}}}{\sin{\theta}}}={{\sin}^{{-{1}}}{\left(-{\frac{{{3}}}{{{4}}}}\right)}}$$
$$\displaystyle\theta={{\sin}^{{-{1}}}{\left(-{\frac{{{3}}}{{{4}}}}\right)}}$$
$$\displaystyle={4},{5.435}$$
The solution repeats value of the equation at every length of $$\displaystyle\pi$$ in the interval $$\displaystyle{\left[{0},{2}\pi\right]}$$.
We will get all solutions of the equation by adding integer multiples of $$\displaystyle\pi$$ to these solutions:
$$\displaystyle\theta={4}+{2}{k}\pi$$
$$\displaystyle\theta={5.435}+{2}{k}\pi$$
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 $$\displaystyle{\left[{0},{2}\pi\right]}$$
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 $$\displaystyle\theta={\frac{{\pi}}{{{2}}}}+{k}\pi,\theta={4}+{2}{k}\pi$$ and $$\displaystyle\theta={5.435}+{2}{k}\pi$$
Conclusion:
Hence, the solutions of the trigonometry equation $$\displaystyle{4}{\cos{\theta}}{\sin{\theta}}+{3}{\cos{\theta}}={0}$$ are $$\displaystyle\theta={\frac{{\pi}}{{{2}}}}+{k}\pi,\theta={4}+{2}{k}\pi$$ and $$\displaystyle\theta={5.435}+{2}{k}\pi$$