Solve for ? tan(2x)=-1

$\tan (2 x) = - 1$

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

$2 x = \arctan (- 1)$

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

$2 x = - \frac{π}{4}$

Divide each term by $2$ and simplify.

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

$x = - \frac{π}{4} \cdot \frac{1}{2}$

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

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

$x = - \frac{π}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = - \frac{π}{8}$

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

$2 x = - \frac{π}{4} - π$

Simplify the expression to find the second solution.

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To write $- \frac{- π}{1}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$- \frac{π}{4} + \frac{- π \cdot 4}{1 \cdot 4}$

Multiply $4$ by $1$ .

$- \frac{π}{4} + \frac{- π \cdot 4}{4}$

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

Combine the numerators over the common denominator.

$2 x = \frac{- π - π \cdot 4}{4}$

Multiply $4$ by $- 1$ .

$2 x = \frac{- π - 4 π}{4}$

Subtract $4 π$ from $- π$ .

$2 x = \frac{- 5 π}{4}$

Divide each term by $2$ and simplify.

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

$x = \frac{\frac{- 5 π}{4}}{2}$

Simplify $\frac{\frac{- 5 π}{4}}{2}$ .

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Multiply the numerator by the reciprocal of the denominator.

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

Move the negative in front of the fraction.

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

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

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

$x = - \frac{5 π}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = - \frac{5 π}{8}$

Add $2 π$ to $- \frac{5 π}{8}$ .

$x = - \frac{5 π}{8} + 2 π$

The resulting angle of $\frac{11 π}{8}$ is positive and coterminal with $- \frac{5 π}{8}$ .

$x = \frac{11 π}{8}$

Find the period.

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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 $2$ .

$\frac{π}{2}$

Add $\frac{π}{2}$ to every negative angle to get positive angles.

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Add $\frac{π}{2}$ to $- \frac{π}{8}$ to find the positive angle.

$- \frac{π}{8} + \frac{π}{2}$

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{π}{2} \cdot \frac{4}{4} - \frac{π}{8}$

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$\frac{π \cdot 4}{2 \cdot 4} - \frac{π}{8}$

Multiply $2$ by $4$ .

$\frac{π \cdot 4}{8} - \frac{π}{8}$

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{8}$

Simplify the numerator.

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Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{8}$

Subtract $π$ from $4 π$ .

$\frac{3 π}{8}$

List the new angles.

$x = \frac{3 π}{8}$

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

$x = \frac{3 π}{8} + \frac{π n}{2}, \frac{11 π}{8} + \frac{π n}{2}$ , for any integer $n$

Consolidate the answers.

$x = \frac{3 π}{8} + \frac{π n}{2}$ , for any integer $n$

Solve for ? tan(2x)=-1

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