Question

solve 7 sin(5x) = 2 for the two smallest positive solutions a and b, with a < b
a =
b =
give your answers accurate to at least two decimal places.
add work
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Explanation:

Step1: Isolate the sine function

Given \(7\sin(5x) = 2\), divide both sides by \(7\) to get \(\sin(5x)=\frac{2}{7}\).

Step2: Use the inverse - sine function

We know that if \(\sin\theta = k\), then \(\theta=\sin^{- 1}(k)+2n\pi\) or \(\theta=\pi-\sin^{-1}(k)+2n\pi\), where \(n\in\mathbb{Z}\).
For \(\sin(5x)=\frac{2}{7}\), we have \(5x=\sin^{-1}(\frac{2}{7}) + 2n\pi\) or \(5x=\pi-\sin^{-1}(\frac{2}{7})+2n\pi\).
First, find \(\sin^{-1}(\frac{2}{7})\approx0.29\) (using a calculator).

Case 1: \(5x=\sin^{-1}(\frac{2}{7})+2n\pi\)

When \(n = 0\), \(5x=\sin^{-1}(\frac{2}{7})\approx0.29\), then \(x=\frac{\sin^{-1}(\frac{2}{7})}{5}\approx\frac{0.29}{5}=0.06\).

Case 2: \(5x=\pi-\sin^{-1}(\frac{2}{7})+2n\pi\)

When \(n = 0\), \(5x=\pi-\sin^{-1}(\frac{2}{7})\approx3.14 - 0.29=2.85\), then \(x=\frac{\pi-\sin^{-1}(\frac{2}{7})}{5}\approx\frac{2.85}{5}=0.57\).

Answer:

\(A = 0.06\), \(B = 0.57\)