Solve for ? tan(x)=-1

$\tan (x) = - 1$

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

$x = \arctan (- 1)$

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

$x = - \frac{π}{4}$

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.

$x = - \frac{π}{4} - π$

Simplify the expression to find the second solution.

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Simplify $- \frac{π}{4} - π$ .

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

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

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

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Combine.

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

Multiply $4$ by $1$ .

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

Combine the numerators over the common denominator.

$x = \frac{- π - π \cdot 4}{4}$

Simplify the numerator.

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Multiply $4$ by $- 1$ .

$x = \frac{- π - 4 π}{4}$

Subtract $4 π$ from $- π$ .

$x = \frac{- 5 π}{4}$

Move the negative in front of the fraction.

$x = - \frac{5 π}{4}$

Add $2 π$ to $- \frac{5 π}{4}$ .

$x = - \frac{5 π}{4} + 2 π$

The resulting angle of $\frac{3 π}{4}$ is positive and coterminal with $- \frac{5 π}{4}$ .

$x = \frac{3 π}{4}$

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 $1$ in the formula for period.

$\frac{π}{| 1 |}$

Solve the equation.

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

$\frac{π}{1}$

Divide $π$ by $1$ .

$π$

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

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

$- \frac{π}{4} + π$

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

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

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

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Combine.

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

Multiply $4$ by $1$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Simplify the numerator.

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Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

List the new angles.

$x = \frac{3 π}{4}$

The period of the $\tan (x)$ 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 $n$

Consolidate the answers.

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

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