Question

solving trigonometric equations v involving invers
solving trigonometric equation v
solving trigor c equations involving inverse trigonometric functions.
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use inverse trig functions to find the smallest two positive solutions of $\sin(6.5x + 7.5)=0.7403$ on $0,2\pi)$. separate the two solutions with a comma, and round to four decimal places.

Explanation:

Step1: Apply inverse sine function

Given \(\sin(6.5x + 7.5)=0.7403\). Using the inverse - sine function \(y=\sin^{-1}(x)\), we have \(6.5x+7.5=\sin^{-1}(0.7403)+2k\pi\) or \(6.5x + 7.5=\pi-\sin^{-1}(0.7403)+2k\pi\), where \(k\in\mathbb{Z}\).

First, find \(\sin^{-1}(0.7403)\approx0.8323\) (using a calculator in radian mode).

Step2: Solve for \(x\) in the first case (\(6.5x+7.5=\sin^{-1}(0.7403)+2k\pi\))

When \(k = 0\):

$$
LATEXBLOCK0
$$

Since \(x\in[0,2\pi]\approx[0,6.2832]\), this solution is rejected.

When \(k = 1\):

$$
LATEXBLOCK1
$$

Rejected.

When \(k = 2\):

$$
LATEXBLOCK2
$$

Step3: Solve for \(x\) in the second case (\(6.5x + 7.5=\pi-\sin^{-1}(0.7403)+2k\pi\))

First, \(\pi-\sin^{-1}(0.7403)\approx3.1416 - 0.8323=2.3093\)

When \(k = 0\):

$$
LATEXBLOCK3
$$

Rejected.

When \(k = 1\):

$$
LATEXBLOCK4
$$

When \(k = 2\):

$$
LATEXBLOCK5
$$

Answer:

\(0.1681,0.9075\)