Question

solve the equation for exact solutions over the interval 0, 2π). 10 csc²( x / 2 ) = 5 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 π 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

Divide both sides of \(10\csc^{2}\frac{x}{2} = 5\) by \(10\).
\(\csc^{2}\frac{x}{2}=\frac{5}{10}=\frac{1}{2}\)
Since \(\csc^{2}\theta=\frac{1}{\sin^{2}\theta}\), then \(\frac{1}{\sin^{2}\frac{x}{2}}=\frac{1}{2}\), so \(\sin^{2}\frac{x}{2} = 2\). But \(- 1\leqslant\sin\alpha\leqslant1\) for any real - number \(\alpha\), and \(\sin^{2}\frac{x}{2}=2\) implies \(\sin\frac{x}{2}=\pm\sqrt{2}\). Since \(\sqrt{2}\approx1.414>1\), there is no real - number solution for \(\sin\frac{x}{2}=\pm\sqrt{2}\).

Answer:

B. The solution is the empty set.