Solve for ? sin(3x)=-1

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Trigonometry

$\sin (3 x) = - 1$

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

$3 x = \arcsin (- 1)$

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

$3 x = - \frac{π}{2}$

Divide each term by $3$ and simplify.

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Divide each term in $3 x = - \frac{π}{2}$ by $3$ .

$\frac{3 x}{3} = - \frac{π}{2} \cdot \frac{1}{3}$

Cancel the common factor of $3$ .

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Cancel the common factor.

$\frac{3 x}{3} = - \frac{π}{2} \cdot \frac{1}{3}$

Divide $x$ by $1$ .

$x = - \frac{π}{2} \cdot \frac{1}{3}$

Multiply $- \frac{π}{2} \cdot \frac{1}{3}$ .

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Multiply $\frac{1}{3}$ and $\frac{π}{2}$ .

$x = - \frac{π}{3 \cdot 2}$

Multiply $3$ by $2$ .

$x = - \frac{π}{6}$

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.

$3 x = 2 π + \frac{π}{2} + π$

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 $\frac{2}{2}$ .

$3 x = \frac{2 π}{1} \cdot \frac{2}{2} + \frac{π}{2} + π$

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 $1$ .

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

$3 x = \frac{2 π \cdot 2}{2} + \frac{π}{2} + π$

Combine the numerators over the common denominator.

$3 x = \frac{2 π \cdot 2 + π}{2} + π$

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

$3 x = \frac{2 π \cdot 2 + π}{2} + \frac{π}{1} \cdot \frac{2}{2}$

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 $1$ .

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

$3 x = \frac{2 π \cdot 2 + π}{2} + \frac{π \cdot 2}{2}$

Combine the numerators over the common denominator.

$3 x = \frac{2 π \cdot 2 + π + π \cdot 2}{2}$

Add $2 π \cdot 2$ and $π \cdot 2$ .

$3 x = \frac{3 π \cdot 2 + π}{2}$

Multiply $2$ by $3$ .

$3 x = \frac{6 π + π}{2}$

Add $6 π$ and $π$ .

$3 x = \frac{7 π}{2}$

Divide each term by $3$ and simplify.

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Divide each term in $3 x = \frac{7 π}{2}$ by $3$ .

$\frac{3 x}{3} = \frac{7 π}{2} \cdot \frac{1}{3}$

Cancel the common factor of $3$ .

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Cancel the common factor.

$\frac{3 x}{3} = \frac{7 π}{2} \cdot \frac{1}{3}$

Divide $x$ by $1$ .

$x = \frac{7 π}{2} \cdot \frac{1}{3}$

Multiply $\frac{7 π}{2} \cdot \frac{1}{3}$ .

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Multiply $\frac{7 π}{2}$ and $\frac{1}{3}$ .

$x = \frac{7 π}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{7 π}{6}$

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 $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

Add $\frac{2 π}{3}$ to every negative angle to get positive angles.

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Add $\frac{2 π}{3}$ to $- \frac{π}{6}$ to find the positive angle.

$- \frac{π}{6} + \frac{2 π}{3}$

To write $\frac{2 π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2 π}{3} \cdot \frac{2}{2} - \frac{π}{6}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$\frac{2 π \cdot 2}{3 \cdot 2} - \frac{π}{6}$

Multiply $3$ by $2$ .

$\frac{2 π \cdot 2}{6} - \frac{π}{6}$

Combine the numerators over the common denominator.

$\frac{2 π \cdot 2 - π}{6}$

Simplify the numerator.

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Multiply $2$ by $2$ .

$\frac{4 π - π}{6}$

Subtract $π$ from $4 π$ .

$\frac{3 π}{6}$

Cancel the common factor of $3$ and $6$ .

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Factor $3$ out of $3 π$ .

$\frac{3 (π)}{6}$

Cancel the common factors.

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Factor $3$ out of $6$ .

$\frac{3 π}{3 \cdot 2}$

Cancel the common factor.

$\frac{3 π}{3 \cdot 2}$

Rewrite the expression.

$\frac{π}{2}$

List the new angles.

$x = \frac{π}{2}$

The period of the $\sin (3 x)$ function is $\frac{2 π}{3}$ so values will repeat every $\frac{2 π}{3}$ radians in both directions.

$x = \frac{π}{2} + \frac{2 π n}{3}, \frac{7 π}{6} + \frac{2 π n}{3}$ , for any integer $n$

Consolidate the answers.

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

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