Question

solve the equation for exact solutions over the interval $0,2\pi)$. $9\sec^{2}\frac{x}{7}=3$ select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the solution set is \\{\\}. (type an exact answer, using $\pi$ as needed. type your answer in radians. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) b. the solution is the empty set.

Explanation:

Step1: Simplify the equation

Given \(9\sec^{2}\frac{x}{7} = 3\), divide both sides by \(9\):
\(\sec^{2}\frac{x}{7}=\frac{3}{9}=\frac{1}{3}\)
Since \(\sec^{2}\theta=\frac{1}{\cos^{2}\theta}\), we have \(\frac{1}{\cos^{2}\frac{x}{7}}=\frac{1}{3}\), then \(\cos^{2}\frac{x}{7}=3\). But \(- 1\leqslant\cos\alpha\leqslant1\) for any real \(\alpha\), and \(\cos^{2}\frac{x}{7}=3\) implies \(\cos\frac{x}{7}=\pm\sqrt{3}\). Since \(\sqrt{3}\approx1.732>1\), there is no real - value of \(\frac{x}{7}\) such that \(\cos\frac{x}{7}=\pm\sqrt{3}\).

Answer:

B. The solution is the empty set.