Trigonometry Examples

f (x) = 2 tan (3 x + π)

$f (x) = 2 \tan (3 x + π)$

Step 1

Set the argument in

tan (3 x + π)

$\tan (3 x + π)$ equal to

\frac{π}{2} + π n

$\frac{π}{2} + π n$ to find where the expression is undefined.

3 x + π = \frac{π}{2} + π n

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

n

$n$

Step 2

Solve for

x

$x$ .

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Step 2.1

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 2.1.1

Subtract

π

$π$ from both sides of the equation.

3 x = \frac{π}{2} + π n - π

$3 x = \frac{π}{2} + π n - π$

Step 2.1.2

To write

- π

$- π$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 2.1.3

Combine

- π

$- π$ and

\frac{2}{2}

$\frac{2}{2}$ .

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

$3 x = π n + \frac{π - π \cdot 2}{2}$

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

$2$ by

- 1

$- 1$ .

3 x = π n + \frac{π - 2 π}{2}

$3 x = π n + \frac{π - 2 π}{2}$

Step 2.1.5.1.2

Subtract

2 π

$2 π$ from

π

$π$ .

3 x = π n + \frac{- π}{2}

$3 x = π n + \frac{- π}{2}$

3 x = π n + \frac{- π}{2}

$3 x = π n + \frac{- π}{2}$

Step 2.1.5.2

Move the negative in front of the fraction.

3 x = π n - \frac{π}{2}

$3 x = π n - \frac{π}{2}$

3 x = π n - \frac{π}{2}

$3 x = π n - \frac{π}{2}$

3 x = π n - \frac{π}{2}

$3 x = π n - \frac{π}{2}$

Step 2.2

Divide each term in

3 x = π n - \frac{π}{2}

$3 x = π n - \frac{π}{2}$ by

3

$3$ and simplify.

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Step 2.2.1

Divide each term in

3 x = π n - \frac{π}{2}

$3 x = π n - \frac{π}{2}$ by

3

$3$ .

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

$3$ .

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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