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

Verified step by step guidance

1. Step 1: Start by setting the equation $\tan (3x)=\frac{\sqrt{3}}{3}$.
2. Step 2: Recognize that $\tan (\theta )=\frac{\sqrt{3}}{3}$ corresponds to an angle $\theta$ where $\tan (\theta )=\tan (\frac{\pi }{6})$ or $\tan (\theta )=\tan (\frac{7\pi }{6})$ within the interval $[0,2\pi )$.
3. Step 3: Set $3x=\frac{\pi }{6}+k\pi$ and $3x=\frac{7\pi }{6}+k\pi$ for integer values of $k$, since the tangent function has a period of $\pi$.
4. Step 4: Solve for $x$ in each equation: $x=\frac{\pi }{18}+\frac{k\pi }{3}$ and $x=\frac{7\pi }{18}+\frac{k\pi }{3}$.
5. Step 5: Determine the values of $x$ that fall within the interval $[0,2\pi )$ by substituting different integer values for $k$ and checking the resulting $x$ values.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to ratios of sides in right triangles. Understanding these functions is essential for solving equations involving angles, especially when dealing with multiple angles, as they can exhibit periodic behavior and specific values at key angles.

Multiple Angle Formulas

Multiple angle formulas allow us to express trigonometric functions of multiple angles in terms of single angles. For example, the tangent of a multiple angle can be expressed using the tangent of the angle itself. This is crucial for simplifying and solving equations like tan(3x) in the given problem.

Interval Notation

Interval notation specifies the range of values for which a solution is valid. In this case, the interval [0, 2π) indicates that solutions must be found within one full rotation of the unit circle, which is important for determining all possible angles that satisfy the equation within the specified range.