Question

solve the equation for x, where x is restricted to the given interval. y = 4 tan 2x, for x in (-\frac{\pi}{4}, \frac{\pi}{4}). x = (use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Isolate the tangent function

Divide both sides of the equation \(y = 4\tan2x\) by \(4\).
\(\frac{y}{4}=\tan2x\)

Step2: Apply the inverse tangent function

Since \(x\in(-\frac{\pi}{4},\frac{\pi}{4})\), then \(2x\in(-\frac{\pi}{2},\frac{\pi}{2})\). The inverse - tangent function \(y = \tan^{-1}u\) is the inverse of \(y=\tan u\) for \(u\in(-\frac{\pi}{2},\frac{\pi}{2})\).
If \(\frac{y}{4}=\tan2x\), then \(2x=\tan^{-1}(\frac{y}{4})\)

Step3: Solve for \(x\)

Divide both sides of the equation \(2x=\tan^{-1}(\frac{y}{4})\) by \(2\).
\(x = \frac{1}{2}\tan^{-1}(\frac{y}{4})\)

Answer:

\(x=\frac{1}{2}\tan^{-1}(\frac{y}{4})\)