Precalculus Examples

$6 \tan (x) + 4 = 5 (1 + \tan (x))$

Step 1

Simplify $5 (1 + \tan (x))$ .

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Step 1.1

Rewrite.

$6 \tan (x) + 4 = 0 + 0 + 5 (1 + \tan (x))$

Step 1.2

Simplify by adding zeros.

$6 \tan (x) + 4 = 5 (1 + \tan (x))$

Step 1.3

Apply the distributive property.

$6 \tan (x) + 4 = 5 \cdot 1 + 5 \tan (x)$

Step 1.4

Multiply $5$ by $1$ .

$6 \tan (x) + 4 = 5 + 5 \tan (x)$

Step 2

Move all terms containing $\tan (x)$ to the left side of the equation.

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Step 2.1

Subtract $5 \tan (x)$ from both sides of the equation.

$6 \tan (x) + 4 - 5 \tan (x) = 5$

Step 2.2

Subtract $5 \tan (x)$ from $6 \tan (x)$ .

$\tan (x) + 4 = 5$

Step 3

Move all terms not containing $\tan (x)$ to the right side of the equation.

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Step 3.1

Subtract $4$ from both sides of the equation.

$\tan (x) = 5 - 4$

Step 3.2

Subtract $4$ from $5$ .

$\tan (x) = 1$

Step 4

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

$x = \arctan (1)$

Step 5

Simplify the right side.

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Step 5.1

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 6

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.

$x = π + \frac{π}{4}$

Step 7

Simplify $π + \frac{π}{4}$ .

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Step 7.1

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 7.2

Combine fractions.

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Step 7.2.1

Combine $π$ and $\frac{4}{4}$ .

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

Step 7.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Step 7.3

Simplify the numerator.

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Step 7.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π + π}{4}$

Step 7.3.2

Add $4 π$ and $π$ .

$x = \frac{5 π}{4}$

Step 8

Find the period of $\tan (x)$ .

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Step 8.1

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

$\frac{π}{| b |}$

Step 8.2

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

$\frac{π}{| 1 |}$

Step 8.3

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

$\frac{π}{1}$

Step 8.4

Divide $π$ by $1$ .

$π$

Step 9

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

$x = \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer $n$

Step 10

Consolidate the answers.

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