Solve for ? sin(2x)+1=0

$\sin (2 x) + 1 = 0$

Subtract $1$ from both sides of the equation.

$\sin (2 x) = - 1$

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

$2 x = \arcsin (- 1)$

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

$2 x = - \frac{π}{2}$

Divide each term by $2$ and simplify.

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

$x = - \frac{π}{2} \cdot \frac{1}{2}$

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

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

$x = - \frac{π}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = - \frac{π}{4}$

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.

$2 x = 2 π + \frac{π}{2} + π$

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 $\frac{2}{2}$ .

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

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 $1$ .

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

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

Combine the numerators over the common denominator.

$2 x = \frac{2 π \cdot 2 + π}{2} + π$

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

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

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 $1$ .

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

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

Combine the numerators over the common denominator.

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

Add $2 π \cdot 2$ and $π \cdot 2$ .

$2 x = \frac{3 π \cdot 2 + π}{2}$

Multiply $2$ by $3$ .

$2 x = \frac{6 π + π}{2}$

Add $6 π$ and $π$ .

$2 x = \frac{7 π}{2}$

Divide each term by $2$ and simplify.

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

$x = \frac{7 π}{2} \cdot \frac{1}{2}$

Multiply $\frac{7 π}{2} \cdot \frac{1}{2}$ .

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Multiply $\frac{7 π}{2}$ and $\frac{1}{2}$ .

$x = \frac{7 π}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{7 π}{4}$

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 $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Solve the equation.

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

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 π}{2}$

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

$x = \frac{3 π}{4} + π n, \frac{7 π}{4} + π n$ , for any integer $n$

Consolidate the answers.

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

Solve for ? sin(2x)+1=0

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