Solve over the Interval tan(x/3)=( square root of 3)/3 , [0,2pi)

Solution:

Step 1:

Isolate $x$ by applying the arctan to both sides of the original equation: $\frac{x}{3} = \arctan\left(\frac{\sqrt{3}}{3}\right)$.

Step 2:

Identify the exact value for the arctan expression.

Step 2.1:

Recognize that $\arctan\left(\frac{\sqrt{3}}{3}\right)$ equals $\frac{\pi}{6}$, leading to $\frac{x}{3} = \frac{\pi}{6}$.

Step 3:

Eliminate the fraction by multiplying by $3$: $3 \cdot \frac{x}{3} = 3 \cdot \frac{\pi}{6}$.

Step 4:

Simplify the equation.

Step 4.1:

Remove the common factor on the left.

Step 4.1.1:

Cancel out the $3$: $\cancel{3} \cdot \frac{x}{\cancel{3}} = 3 \cdot \frac{\pi}{6}$.

Step 4.1.2:

Express $x$ as $x = 3 \cdot \frac{\pi}{6}$.

Step 4.2:

Simplify the right side.

Step 4.2.1:

Reduce the fraction by dividing by $3$: $x = \frac{\pi}{2}$.

Step 5:

Since tan is positive in the first and third quadrants, find the second solution by adding $\pi$ to the reference angle: $\frac{x}{3} = \pi + \frac{\pi}{6}$.

Step 6:

Solve for the second value of $x$.

Step 6.1:

Multiply by $3$: $3 \cdot \frac{x}{3} = 3 \cdot (\pi + \frac{\pi}{6})$.

Step 6.2:

Simplify the equation.

Step 6.2.1:

Cancel the $3$ on the left.

Step 6.2.2:

Simplify the right side.

Step 6.2.2.1:

Express $\pi$ with a common denominator: $x = 3 \cdot (\pi \cdot \frac{6}{6} + \frac{\pi}{6})$.

Step 6.2.2.2:

Combine like terms: $x = 3 \cdot \frac{6\pi + \pi}{6}$.

Step 6.2.2.3:

Reduce the fraction: $x = \frac{7\pi}{2}$.

Step 7:

Determine the period of $\tan\left(\frac{x}{3}\right)$.

Step 7.1:

Use the period formula: $\frac{\pi}{|b|}$.

Step 7.2:

Substitute $b$ with $\frac{1}{3}$: $\frac{\pi}{|\frac{1}{3}|}$.

Step 7.3:

Since $\frac{1}{3}$ is positive, remove the absolute value: $\frac{\pi}{\frac{1}{3}}$.

Step 7.4:

Multiply by the reciprocal of the denominator: $\pi \cdot 3$.

Step 7.5:

Rearrange to $3\pi$.

Step 8:

Account for the period to find all solutions: $x = \frac{\pi}{2} + 3\pi n$ and $x = \frac{7\pi}{2} + 3\pi n$, where $n$ is any integer.

Step 9:

Combine the solutions into one expression: $x = \frac{\pi}{2} + 3\pi n$, where $n$ is any integer.

Step 10:

Verify if the solutions fall within the interval $[0, 2\pi)$.

Step 10.1:

Substitute $n = 0$: $\frac{\pi}{2} + 3\pi \cdot 0$.

Step 10.2:

Simplify the expression.

Step 10.2.1:

Calculate $3\pi \cdot 0$: $\frac{\pi}{2} + 0$.

Step 10.2.2:

Add the results: $\frac{\pi}{2}$.

Step 10.3:

Confirm that $\frac{\pi}{2}$ is within the interval $[0, 2\pi)$: $x = \frac{\pi}{2}$.

Knowledge Notes:

The problem involves solving a trigonometric equation over a specified interval. Here are the relevant knowledge points:

1. Inverse Trigonometric Functions: These functions, such as $\arctan(x)$, are used to find the angle whose tangent is $x$. They are the inverse operations of the trigonometric functions.

2. Exact Trigonometric Values: Certain angles have exact values for their trigonometric functions, which are often derived from the unit circle or special right triangles.

3. Simplifying Expressions: This involves canceling common factors, combining like terms, and reducing fractions to their simplest form.

4. Periodicity of Trigonometric Functions: Trigonometric functions repeat their values in regular intervals called periods. For the tangent function, the period is $\pi$.

5. Solving Trigonometric Equations: This may involve using algebraic techniques to isolate the variable and considering the periodic nature of the trigonometric functions to find all possible solutions within a given interval.

6. Quadrant Analysis: Since trigonometric functions have different signs in different quadrants, it's important to consider the reference angle and the sign of the function to find all solutions.

7. Interval Notation: The interval $[0, 2\pi)$ includes all real numbers from $0$ to $2\pi$, including $0$ but not $2\pi$. This is important when determining which solutions are valid within the given interval.