16. 0/1 points details my notes larpcalclim4hs 5.3.049. use a graphing utility to approximate (to three…

16. 0/1 points details my notes larpcalclim4hs 5.3.049. use a graphing utility to approximate (to three decimal places) the solutions of the equation in the interval 0, 2π). (enter your answers as a comma - separated list.) 5 sin(x) + 3 = 0 x = 5.693 x need help? read it submit answer

16. 0/1 points details my notes larpcalclim4hs 5.3.049. use a graphing utility to approximate (to three decimal places) the solutions of the equation in the interval 0, 2π). (enter your answers as a comma - separated list.) 5 sin(x) + 3 = 0 x = 5.693 x need help? read it submit answer

Answer

Explanation:

Step1: Isolate sin(x)

First, solve the equation $5\sin(x)+3 = 0$ for $\sin(x)$. Subtract 3 from both sides: $5\sin(x)=- 3$, then $\sin(x)=-\frac{3}{5}=-0.6$.

Step2: Find reference - angle

The reference - angle $\alpha$ is given by $\sin\alpha = 0.6$. Using the inverse - sine function, $\alpha=\sin^{-1}(0.6)\approx0.644$ radians.

Step3: Find solutions in [0, 2π)

Since $\sin(x)<0$, the solutions are in the third and fourth quadrants. In the third quadrant, $x=\pi+\alpha=\pi + 0.644\approx3.785$. In the fourth quadrant, $x = 2\pi-\alpha=2\pi-0.644\approx5.639$.

Answer:

$3.785,5.639$