Finite Math Examples

$y = \tan (x + \frac{π}{4})$

Step 1

Set $\tan (x + \frac{π}{4})$ equal to $0$ .

$\tan (x + \frac{π}{4}) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Take the inverse tangent of both sides of the equation to extract $x$ from inside the tangent.

$x + \frac{π}{4} = \arctan (0)$

Step 2.2

Simplify the right side.

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Step 2.2.1

The exact value of $\arctan (0)$ is $0$ .

$x + \frac{π}{4} = 0$

Step 2.3

Subtract $\frac{π}{4}$ from both sides of the equation.

$x = - \frac{π}{4}$

Step 2.4

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x + \frac{π}{4} = π + 0$

Step 2.5

Solve for $x$ .

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Step 2.5.1

Add $π$ and $0$ .

$x + \frac{π}{4} = π$

Step 2.5.2

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

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Step 2.5.2.1

Subtract $\frac{π}{4}$ from both sides of the equation.

$x = π - \frac{π}{4}$

Step 2.5.2.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = π \cdot \frac{4}{4} - \frac{π}{4}$

Step 2.5.2.3

Combine $π$ and $\frac{4}{4}$ .

$x = \frac{π \cdot 4}{4} - \frac{π}{4}$

Step 2.5.2.4

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 - π}{4}$

Step 2.5.2.5

Simplify the numerator.

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Step 2.5.2.5.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π - π}{4}$

Step 2.5.2.5.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{4}$

Step 2.6

Find the period of $\tan (x + \frac{π}{4})$ .

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Step 2.6.1

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 2.6.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 2.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 2.6.4

Divide $π$ by $1$ .

$π$

Step 2.7

Add $π$ to every negative angle to get positive angles.

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Step 2.7.1

Add $π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + π$

Step 2.7.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 2.7.3

Combine fractions.

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Step 2.7.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 2.7.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 2.7.4

Simplify the numerator.

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Step 2.7.4.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Step 2.7.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

Step 2.7.5

List the new angles.

$x = \frac{3 π}{4}$

Step 2.8

The period of the $\tan (x + \frac{π}{4})$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 2.9

Consolidate the answers.

$x = \frac{3 π}{4} + π n$ , for any integer $n$

Step 3