Solve each equation for x.
y = 1/2 tan (3x + 2), for x in [-2/3 - π/6, -2/3 + π/6]

Key Concepts

Tangent Function

The tangent function, denoted as tan(x), is a fundamental trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It is periodic with a period of π, meaning it repeats its values every π radians. Understanding the properties of the tangent function, including its asymptotes and behavior, is crucial for solving equations involving tan.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan, are used to find angles when the value of a trigonometric function is known. For example, if y = tan(x), then x = arctan(y). These functions are essential for solving equations where the variable is inside a trigonometric function, allowing us to isolate the angle and find the corresponding x values.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In this context, the interval [-2/3 - π/6, -2/3 + π/6] specifies the domain within which we are looking for solutions for x. Understanding how to interpret and work within specified intervals is important for determining valid solutions to trigonometric equations.