Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). tan 3x = (√3)/3

Verified step by step guidance

1. Start by writing down the given equation: \(\tan 3x = \frac{\sqrt{3}}{3}\).
2. Recall the values of tangent for common angles. Since \(\tan \theta = \frac{\sqrt{3}}{3}\), identify the reference angle \(\theta\) where this is true. This corresponds to \(\theta = \frac{\pi}{6}\) (or 30 degrees).
3. Set up the general solution for \$3x$ using the periodicity of the tangent function, which has period \(\pi\). So, \(3x = \frac{\pi}{6} + k\pi\), where $k$ is any integer.
4. Solve for \(x\) by dividing both sides by 3: \(x = \frac{\pi}{18} + \frac{k\pi}{3}\).
5. Find all values of \(x\) in the interval \([0, 2\pi)\) by substituting integer values of \(k\) such that \(x\) remains within this interval.

Key Concepts

Multiple-Angle Trigonometric Equations

These are equations where the trigonometric function's argument is a multiple of the variable, such as tan(3x). Solving them requires understanding how to handle the periodicity and multiple solutions within a given interval, often by dividing the interval accordingly.

Properties and Periodicity of the Tangent Function

The tangent function has a period of π, meaning tan(θ) = tan(θ + nπ) for any integer n. This periodicity is crucial when solving equations like tan(3x) = value, as it helps find all solutions within the specified interval by considering all possible angle shifts.

Solving Trigonometric Equations on a Restricted Interval

When solving trig equations on [0, 2π), it is important to find all solutions within this range. For multiple-angle equations, the variable's domain is adjusted accordingly (e.g., 3x in [0, 6π)), and solutions are then translated back to x by dividing by the multiple.