Question

solve the equation for exact solutions - 4 arcsin x = π solve the equation for exact solutions. select the correct choice below and, if necessary, fill in the answer box within your choice. a. the solution set is (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) b. the solution is the empty set

Explanation:

Step1: Isolate the inverse sine function

Divide both sides of the equation \(-4\arcsin x=\pi\) by \(-4\).
\(\arcsin x =-\frac{\pi}{4}\)

Step2: Use the definition of the inverse sine function

By the definition of \(y = \arcsin x\) (where \(-\frac{\pi}{2}\leq y\leq\frac{\pi}{2}\) and \(- 1\leq x\leq1\)), if \(\arcsin x=-\frac{\pi}{4}\), then \(x=\sin(-\frac{\pi}{4})\)
Since \(\sin(-\alpha)=-\sin\alpha\), we have \(x =-\sin(\frac{\pi}{4})\)
And \(\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\), so \(x =-\frac{\sqrt{2}}{2}\)

Answer:

A. The solution set is \(-\frac{\sqrt{2}}{2}\)