chapter 6 quiz\n170 points possible answered: 15/17\nquestion 16\nsolve ( 5 sin left( \frac { pi } { 4 } x…

Explanation:

Step1: Isolate the sine function

Divide both sides of the equation (5\sin\left(\frac{\pi}{4}x\right)=2) by (5). (\sin\left(\frac{\pi}{4}x\right)=\frac{2}{5} = 0.4)

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}). Let (\theta=\frac{\pi}{4}x). Then (\frac{\pi}{4}x=\sin^{-1}(0.4)+2k\pi) or (\frac{\pi}{4}x=\pi-\sin^{-1}(0.4)+2k\pi) First, find (\sin^{-1}(0.4)\approx0.4115) (in radians) For (\frac{\pi}{4}x=\sin^{-1}(0.4)+2k\pi): (x=\frac{4}{\pi}(\sin^{-1}(0.4)+2k\pi)) For (\frac{\pi}{4}x=\pi - \sin^{-1}(0.4)+2k\pi): (x=\frac{4}{\pi}(\pi-\sin^{-1}(0.4)+2k\pi))

Step3: Find the four smallest positive solutions

When (k = 0) for (x=\frac{4}{\pi}(\sin^{-1}(0.4)+2k\pi)): (x_1=\frac{4}{\pi}\times0.4115\approx0.52) When (k = 0) for (x=\frac{4}{\pi}(\pi-\sin^{-1}(0.4)+2k\pi)): (x_2=\frac{4}{\pi}(\pi - 0.4115)\approx\frac{4}{\pi}\times2.7301\approx3.48) When (k = 1) for (x=\frac{4}{\pi}(\sin^{-1}(0.4)+2k\pi)): (x_3=\frac{4}{\pi}(0.4115 + 2\pi)\approx\frac{4}{\pi}(0.4115+6.2832)\approx\frac{4}{\pi}\times6.6947\approx8.52) When (k = 1) for (x=\frac{4}{\pi}(\pi-\sin^{-1}(0.4)+2k\pi)): (x_4=\frac{4}{\pi}(\pi - 0.4115+2\pi)=\frac{4}{\pi}(3\pi - 0.4115)\approx\frac{4}{\pi}(9.4248 - 0.4115)\approx\frac{4}{\pi}\times9.0133\approx11.48)

Answer:

(0.52,3.48,8.52,11.48)