Solve for x -sin(pix)=1

$- \sin (π x) = 1$

Multiply each term in $- \sin (π x) = 1$ by $- 1$

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Multiply each term in $- \sin (π x) = 1$ by $- 1$ .

$(- \sin (π x)) \cdot - 1 = 1 \cdot - 1$

Multiply $- \sin (π x) \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$1 \sin (π x) = 1 \cdot - 1$

Multiply $\sin (π x)$ by $1$ .

$\sin (π x) = 1 \cdot - 1$

Multiply $- 1$ by $1$ .

$\sin (π x) = - 1$

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

$π x = \arcsin (- 1)$

The exact value of $\arcsin (- 1)$ is $- \frac{π}{2}$ .

$π x = - \frac{π}{2}$

Divide each term by $π$ and simplify.

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

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

Cancel the common factor of $π$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

$x = - \frac{π}{2} \cdot \frac{1}{π}$

Simplify $- \frac{π}{2} \cdot \frac{1}{π}$ .

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Cancel the common factor of $π$ .

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Move the leading negative in $- \frac{π}{2}$ into the numerator.

$x = \frac{- π}{2} \cdot \frac{1}{π}$

Factor $π$ out of $- π$ .

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

Cancel the common factor.

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

Rewrite the expression.

$x = \frac{- 1}{2}$

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$π x = 2 π + \frac{π}{2} + π$

Simplify the expression to find the second solution.

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To write $\frac{2 π}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$\frac{2 π \cdot 2}{1 \cdot 2} + \frac{π}{2} + π$

Multiply $2$ by $1$ .

$\frac{2 π \cdot 2}{2} + \frac{π}{2} + π$

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

Combine the numerators over the common denominator.

$π x = \frac{2 π \cdot 2 + π}{2} + π$

To write $\frac{π}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$\frac{2 π \cdot 2 + π}{2} + \frac{π \cdot 2}{1 \cdot 2}$

Multiply $2$ by $1$ .

$\frac{2 π \cdot 2 + π}{2} + \frac{π \cdot 2}{2}$

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

Combine the numerators over the common denominator.

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

Add $2 π \cdot 2$ and $π \cdot 2$ .

$π x = \frac{3 π \cdot 2 + π}{2}$

Multiply $2$ by $3$ .

$π x = \frac{6 π + π}{2}$

Add $6 π$ and $π$ .

$π x = \frac{7 π}{2}$

Divide each term by $π$ and simplify.

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Divide each term in $π x = \frac{7 π}{2}$ by $π$ .

$\frac{π x}{π} = \frac{7 π}{2} \cdot \frac{1}{π}$

Cancel the common factor of $π$ .

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Cancel the common factor.

$\frac{π x}{π} = \frac{7 π}{2} \cdot \frac{1}{π}$

Divide $x$ by $1$ .

$x = \frac{7 π}{2} \cdot \frac{1}{π}$

Cancel the common factor of $π$ .

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Factor $π$ out of $7 π$ .

$x = \frac{π \cdot 7}{2} \cdot \frac{1}{π}$

Cancel the common factor.

$x = \frac{π \cdot 7}{2} \cdot \frac{1}{π}$

Rewrite the expression.

$x = \frac{7}{2}$

Find the period.

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The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| π |}$

Solve the equation.

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$π$ is approximately $3.14159265$ which is positive so remove the absolute value

$\frac{2 π}{π}$

Cancel the common factor of $π$ .

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Cancel the common factor.

$\frac{2 π}{π}$

Divide $2$ by $1$ .

$2$

Add $2$ to every negative angle to get positive angles.

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Add $2$ to $- \frac{1}{2}$ to find the positive angle.

$- \frac{1}{2} + 2$

To write $\frac{2}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2}{1} \cdot \frac{2}{2} - \frac{1}{2}$

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$\frac{2 \cdot 2}{1 \cdot 2} - \frac{1}{2}$

Multiply $2$ by $1$ .

$\frac{2 \cdot 2}{2} - \frac{1}{2}$

Combine the numerators over the common denominator.

$\frac{2 \cdot 2 - 1}{2}$

Simplify the numerator.

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Multiply $2$ by $2$ .

$\frac{4 - 1}{2}$

Subtract $1$ from $4$ .

$\frac{3}{2}$

List the new angles.

$x = \frac{3}{2}$

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

$x = \frac{3}{2} + 2 n, \frac{7}{2} + 2 n$ , for any integer $n$

Consolidate the answers.

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

Solve for x -sin(pix)=1

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