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. Start by isolating the trigonometric function: divide both sides of the equation by 3 to get \( \tan(2x) = \frac{y}{3} \).
2. To solve for \( x \), take the inverse tangent (arctan) of both sides: \( 2x = \arctan\left(\frac{y}{3}\right) \).
3. Solve for \( x \) by dividing both sides by 2: \( x = \frac{1}{2} \arctan\left(\frac{y}{3}\right) \).
4. Consider the periodicity of the tangent function, which has a period of \( \pi \). Since \( 2x \) is involved, the period becomes \( \frac{\pi}{2} \).
5. Ensure that the solution for \( x \) falls within the interval \([-\frac{\pi}{4}, \frac{\pi}{4}]\) by checking if any additional solutions from the periodicity need to be included.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to ratios of sides in right triangles. The tangent function, specifically, is defined as the ratio of the opposite side to the adjacent side. Understanding how these functions behave, including their periodicity and asymptotes, is crucial for solving equations involving them.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles when given a trigonometric ratio. For example, the inverse tangent function (arctan) is used to determine the angle whose tangent is a specific value. This concept is essential when solving equations for x, as it helps to isolate the variable and find its corresponding angle.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In this case, the interval [―π/4, π/4] indicates that x can take any value from -π/4 to π/4, inclusive. Understanding how to interpret and apply these intervals is important for ensuring that solutions to trigonometric equations fall within the specified range.