Find All Complex Solutions tan(2x)-1=0

$\tan (2 x) - 1 = 0$

Step 1

Add $1$ to both sides of the equation.

$\tan (2 x) = 1$

Step 2

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

$2 x = \arctan (1)$

Step 3

Simplify the right side.

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

$2 x = \frac{π}{4}$

Step 4

Divide each term in $2 x = \frac{π}{4}$ by $2$ and simplify.

Divide each term in $2 x = \frac{π}{4}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{π}{4}}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{π}{4}}{2}$

Divide $x$ by $1$ .

$x = \frac{\frac{π}{4}}{2}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{4} \cdot \frac{1}{2}$

Multiply $\frac{π}{4} \cdot \frac{1}{2}$ .

Multiply $\frac{π}{4}$ by $\frac{1}{2}$ .

$x = \frac{π}{4 \cdot 2}$

Multiply $4$ by $2$ .

$x = \frac{π}{8}$

Step 5

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.

$2 x = π + \frac{π}{4}$

Step 6

Solve for $x$ .

Simplify.

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

$2 x = π \cdot \frac{4}{4} + \frac{π}{4}$

Combine $π$ and $\frac{4}{4}$ .

$2 x = \frac{π \cdot 4}{4} + \frac{π}{4}$

Combine the numerators over the common denominator.

$2 x = \frac{π \cdot 4 + π}{4}$

Add $π \cdot 4$ and $π$ .

Reorder $π$ and $4$ .

$2 x = \frac{4 \cdot π + π}{4}$

Add $4 \cdot π$ and $π$ .

$2 x = \frac{5 \cdot π}{4}$

Divide each term in $2 x = \frac{5 \cdot π}{4}$ by $2$ and simplify.

Divide each term in $2 x = \frac{5 \cdot π}{4}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{5 \cdot π}{4}}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{5 \cdot π}{4}}{2}$

Divide $x$ by $1$ .

$x = \frac{\frac{5 \cdot π}{4}}{2}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5 \cdot π}{4} \cdot \frac{1}{2}$

Multiply $\frac{5 π}{4} \cdot \frac{1}{2}$ .

Multiply $\frac{5 π}{4}$ by $\frac{1}{2}$ .

$x = \frac{5 π}{4 \cdot 2}$

Multiply $4$ by $2$ .

$x = \frac{5 π}{8}$

Step 7

Find the period of $\tan (2 x)$ .

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

$\frac{π}{| b |}$

Replace $b$ with $2$ in the formula for period.

$\frac{π}{| 2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{π}{2}$

Step 8

The period of the $\tan (2 x)$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in both directions.

$x = \frac{π}{8} + \frac{π n}{2}, \frac{5 π}{8} + \frac{π n}{2}$ , for any integer $n$

Step 9

Consolidate the answers.

$x = \frac{π}{8} + \frac{π n}{2}$ , for any integer $n$

Find All Complex Solutions tan(2x)-1=0

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