Solve the given equation. (Enter your answers as a comma-separated list. Let $k$ be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.) $7 \cos(\theta) \sin(\theta) + 4 \cos(\theta) = 0$

Full solution

Q. Solve the given equation. (Enter your answers as a comma-separated list. Let

k

be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.)

7 \cos(\theta) \sin(\theta) + 4 \cos(\theta) = 0

Factor out $\cos(\theta)$ : Factor out $\cos(\theta)$ from the equation. $7 \cos(\theta) \sin(\theta) + 4 \cos(\theta) = \cos(\theta)(7 \sin(\theta) + 4) = 0$
Solve for $\theta$ : Set each factor equal to zero and solve for $\theta$ . $\cos(\theta) = 0$ and $7 \sin(\theta) + 4 = 0$
Solve $\cos(\theta):$ Solve $\cos(\theta) = 0.$ $\newline$ $\theta = \frac{\pi}{2} + k\pi,$ where $k$ is any integer.
Solve for $\sin(\theta)$ : Solve $7 \sin(\theta) + 4 = 0$ . $\newline$ $\sin(\theta) = -\frac{4}{7}$
Find $\theta$ : Find $\theta$ for $\sin(\theta) = -\frac{4}{7}$ . $\newline$ $\theta = \arcsin(-\frac{4}{7})$ and $\theta = \pi - \arcsin(-\frac{4}{7})$
Combine solutions: Combine all solutions. $\theta = \frac{\pi}{2} + k\pi$ , $\arcsin\left(-\frac{4}{7}\right)$ , $\pi - \arcsin\left(-\frac{4}{7}\right)$