solve 7 sin(π/3 x) = 4 for the four smallest positive solutions. round answers to 3 decimal places. x =…

Explanation:

Step1: Isolate the sine - function

Divide both sides of the equation $7\sin(\frac{\pi}{3}x)=4$ by 7. We get $\sin(\frac{\pi}{3}x)=\frac{4}{7}\approx0.571$.

Step2: Use the inverse - sine function

We know that if $\sin\theta = a$, then $\theta=\sin^{-1}(a)+2k\pi$ or $\theta=\pi - \sin^{-1}(a)+2k\pi$, where $k\in\mathbb{Z}$. For $\sin(\frac{\pi}{3}x)=0.571$, we have $\frac{\pi}{3}x=\sin^{-1}(0.571)+2k\pi$ or $\frac{\pi}{3}x=\pi-\sin^{-1}(0.571)+2k\pi$. First, find $\sin^{-1}(0.571)\approx0.615$ (in radians).

Step3: Solve for $x$ in the first case

From $\frac{\pi}{3}x = 0.615+2k\pi$, we can solve for $x$: $x=\frac{3(0.615 + 2k\pi)}{\pi}=\frac{1.845}{\pi}+6k$. When $k = 0$, $x_1=\frac{1.845}{\pi}\approx0.587$.

Step4: Solve for $x$ in the second case

From $\frac{\pi}{3}x=\pi - 0.615+2k\pi$, we can solve for $x$: $x=\frac{3(\pi - 0.615+2k\pi)}{\pi}=3-\frac{1.845}{\pi}+6k$. When $k = 0$, $x_2=3 - 0.587 = 2.413$. When $k = 1$ in the first case: $x_3=\frac{1.845}{\pi}+6\approx0.587 + 6=6.587$. When $k = 1$ in the second case: $x_4=3-\frac{1.845}{\pi}+6=9 - 0.587=8.413$.

Answer:

$x = 0.587,2.413,6.587,8.413$