Trigonometry Examples

$f (x) = 2 \tan (3 x + π)$

Step 1

Set the argument in $\tan (3 x + π)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$3 x + π = \frac{π}{2} + π n$ , for any integer $n$

Step 2

Solve for $x$ .

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Step 2.1

Move all terms not containing $x$ to the right side of the equation.

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Step 2.1.1

Subtract $π$ from both sides of the equation.

$3 x = \frac{π}{2} + π n - π$

Step 2.1.2

To write $- π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 x = π n + \frac{π}{2} - π \cdot \frac{2}{2}$

Step 2.1.3

Combine $- π$ and $\frac{2}{2}$ .

$3 x = π n + \frac{π}{2} + \frac{- π \cdot 2}{2}$

Step 2.1.4

Combine the numerators over the common denominator.

$3 x = π n + \frac{π - π \cdot 2}{2}$

Step 2.1.5

Simplify each term.

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Step 2.1.5.1

Simplify the numerator.

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Step 2.1.5.1.1

Multiply $2$ by $- 1$ .

$3 x = π n + \frac{π - 2 π}{2}$

Step 2.1.5.1.2

Subtract $2 π$ from $π$ .

$3 x = π n + \frac{- π}{2}$

Step 2.1.5.2

Move the negative in front of the fraction.

$3 x = π n - \frac{π}{2}$

Step 2.2

Divide each term in $3 x = π n - \frac{π}{2}$ by $3$ and simplify.

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Step 2.2.1

Divide each term in $3 x = π n - \frac{π}{2}$ by $3$ .

$\frac{3 x}{3} = \frac{π n}{3} + \frac{- \frac{π}{2}}{3}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{π n}{3} + \frac{- \frac{π}{2}}{3}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π n}{3} + \frac{- \frac{π}{2}}{3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π n}{3} - \frac{π}{2} \cdot \frac{1}{3}$

Step 2.2.3.1.2

Multiply $- \frac{π}{2} \cdot \frac{1}{3}$ .

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Step 2.2.3.1.2.1

Multiply $\frac{1}{3}$ by $\frac{π}{2}$ .

$x = \frac{π n}{3} - \frac{π}{3 \cdot 2}$

Step 2.2.3.1.2.2

Multiply $3$ by $2$ .

$x = \frac{π n}{3} - \frac{π}{6}$

Step 3

The domain is all values of $x$ that make the expression defined.

Set-Builder Notation:

${x | x \neq \frac{π n}{3} - \frac{π}{6}}$ , for any integer $n$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 5

Determine the domain and range.

Domain: ${x | x \neq \frac{π n}{3} - \frac{π}{6}}$ , for any integer $n$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 6