question 16\nsolve ( 5 sin left( \frac { pi } { 4 } x \right) = 2 ) for the four smallest positive…

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}\times2.73\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}(2.73+2\pi)\approx\frac{4}{\pi}(2.73 + 6.2832)=\frac{4}{\pi}\times9.0132\approx11.48)

Answer:

(x = 0.52,3.48,8.52,11.48)