Solve each equation for x, where x is restricted to the given interval.
y = 3 tan 2x , for x in [―π/4, π/4]

Verified step by step guidance

1. Rewrite the given equation clearly: \(y = 3 \tan(2x)\). To solve for \(x\), we need to isolate \(x\) in terms of \(y\).
2. Divide both sides of the equation by 3 to isolate the tangent function: \(\tan(2x) = \frac{y}{3}\).
3. Apply the inverse tangent (arctangent) function to both sides to solve for \$2x$: \(2x = \arctan\left(\frac{y}{3}\right)\).
4. Divide both sides by 2 to solve for \(x\): \(x = \frac{1}{2} \arctan\left(\frac{y}{3}\right)\).
5. Since \(x\) is restricted to the interval \(\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]\), check that the values of \(x\) obtained from the inverse tangent fall within this interval, considering the periodicity and range of the tangent function.

Key Concepts

Properties of the Tangent Function

The tangent function, tan(θ), is periodic with period π and has vertical asymptotes where cos(θ) = 0. Understanding its behavior, including its range and points of discontinuity, is essential for solving equations involving tangent, especially when restricting the domain.

Solving Trigonometric Equations

Solving equations like y = 3 tan(2x) involves isolating the trigonometric function and using inverse functions to find general solutions. One must consider the periodicity of tangent and apply domain restrictions to identify all valid solutions within the given interval.

Domain and Interval Restrictions

When solving trigonometric equations, restricting the variable x to a specific interval, such as [−π/4, π/4], limits the possible solutions. It is crucial to check which solutions fall within this interval to ensure the answer set is accurate and complete.