Precalculus Examples

$2 \sin (5 x - \frac{1}{12} \cdot π) = - \sqrt{2}$

Step 1

Divide each term in $2 \sin (5 x - \frac{1}{12} \cdot π) = - \sqrt{2}$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 \sin (5 x - \frac{1}{12} \cdot π) = - \sqrt{2}$ by $2$ .

$\frac{2 \sin (5 x - \frac{1}{12} \cdot π)}{2} = \frac{- \sqrt{2}}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 \sin (5 x - \frac{1}{12} \cdot π)}{2} = \frac{- \sqrt{2}}{2}$

Step 1.2.1.2

Divide $\sin (5 x - \frac{1}{12} \cdot π)$ by $1$ .

$\sin (5 x - \frac{1}{12} \cdot π) = \frac{- \sqrt{2}}{2}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

$\sin (5 x - \frac{1}{12} \cdot π) = - \frac{\sqrt{2}}{2}$

Step 2

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

$5 x - \frac{1}{12} \cdot π = \arcsin (- \frac{\sqrt{2}}{2})$

Step 3

Simplify the left side.

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Step 3.1

Combine $π$ and $\frac{1}{12}$ .

$5 x - \frac{π}{12} = \arcsin (- \frac{\sqrt{2}}{2})$

Step 4

Simplify the right side.

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Step 4.1

The exact value of $\arcsin (- \frac{\sqrt{2}}{2})$ is $- \frac{π}{4}$ .

$5 x - \frac{π}{12} = - \frac{π}{4}$

Step 5

Move all terms not containing $x$ to the right side of the equation.

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Step 5.1

Add $\frac{π}{12}$ to both sides of the equation.

$5 x = - \frac{π}{4} + \frac{π}{12}$

Step 5.2

To write $- \frac{π}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$5 x = - \frac{π}{4} \cdot \frac{3}{3} + \frac{π}{12}$

Step 5.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.1

Multiply $\frac{π}{4}$ by $\frac{3}{3}$ .

$5 x = - \frac{π \cdot 3}{4 \cdot 3} + \frac{π}{12}$

Step 5.3.2

Multiply $4$ by $3$ .

$5 x = - \frac{π \cdot 3}{12} + \frac{π}{12}$

Step 5.4

Combine the numerators over the common denominator.

$5 x = \frac{- π \cdot 3 + π}{12}$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Multiply $3$ by $- 1$ .

$5 x = \frac{- 3 π + π}{12}$

Step 5.5.2

Add $- 3 π$ and $π$ .

$5 x = \frac{- 2 π}{12}$

Step 5.6

Cancel the common factor of $- 2$ and $12$ .

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Step 5.6.1

Factor $2$ out of $- 2 π$ .

$5 x = \frac{2 (- π)}{12}$

Step 5.6.2

Cancel the common factors.

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Step 5.6.2.1

Factor $2$ out of $12$ .

$5 x = \frac{2 (- π)}{2 (6)}$

Step 5.6.2.2

Cancel the common factor.

$5 x = \frac{2 (- π)}{2 \cdot 6}$

Step 5.6.2.3

Rewrite the expression.

$5 x = \frac{- π}{6}$

Step 5.7

Move the negative in front of the fraction.

$5 x = - \frac{π}{6}$

Step 6

Divide each term in $5 x = - \frac{π}{6}$ by $5$ and simplify.

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Step 6.1

Divide each term in $5 x = - \frac{π}{6}$ by $5$ .

$\frac{5 x}{5} = \frac{- \frac{π}{6}}{5}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $5$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- \frac{π}{6}}{5}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{6}}{5}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{6} \cdot \frac{1}{5}$

Step 6.3.2

Multiply $- \frac{π}{6} \cdot \frac{1}{5}$ .

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Step 6.3.2.1

Multiply $\frac{1}{5}$ by $\frac{π}{6}$ .

$x = - \frac{π}{5 \cdot 6}$

Step 6.3.2.2

Multiply $5$ by $6$ .

$x = - \frac{π}{30}$

Step 7

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.

$5 x - \frac{π}{12} = 2 π + \frac{π}{4} + π$

Step 8

Simplify the expression to find the second solution.

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Step 8.1

Subtract $2 π$ from $2 π + \frac{π}{4} + π$ .

$5 x - \frac{π}{12} = 2 π + \frac{π}{4} + π - 2 π$

Step 8.2

The resulting angle of $\frac{5 π}{4}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{4} + π$ .

$5 x - \frac{π}{12} = \frac{5 π}{4}$

Step 8.3

Solve for $x$ .

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Step 8.3.1

Move all terms not containing $x$ to the right side of the equation.

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Step 8.3.1.1

Add $\frac{π}{12}$ to both sides of the equation.

$5 x = \frac{5 π}{4} + \frac{π}{12}$

Step 8.3.1.2

To write $\frac{5 π}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$5 x = \frac{5 π}{4} \cdot \frac{3}{3} + \frac{π}{12}$

Step 8.3.1.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.3.1.3.1

Multiply $\frac{5 π}{4}$ by $\frac{3}{3}$ .

$5 x = \frac{5 π \cdot 3}{4 \cdot 3} + \frac{π}{12}$

Step 8.3.1.3.2

Multiply $4$ by $3$ .

$5 x = \frac{5 π \cdot 3}{12} + \frac{π}{12}$

Step 8.3.1.4

Combine the numerators over the common denominator.

$5 x = \frac{5 π \cdot 3 + π}{12}$

Step 8.3.1.5

Simplify the numerator.

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Step 8.3.1.5.1

Multiply $3$ by $5$ .

$5 x = \frac{15 π + π}{12}$

Step 8.3.1.5.2

Add $15 π$ and $π$ .

$5 x = \frac{16 π}{12}$

Step 8.3.1.6

Cancel the common factor of $16$ and $12$ .

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Step 8.3.1.6.1

Factor $4$ out of $16 π$ .

$5 x = \frac{4 (4 π)}{12}$

Step 8.3.1.6.2

Cancel the common factors.

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Step 8.3.1.6.2.1

Factor $4$ out of $12$ .

$5 x = \frac{4 (4 π)}{4 (3)}$

Step 8.3.1.6.2.2

Cancel the common factor.

$5 x = \frac{4 (4 π)}{4 \cdot 3}$

Step 8.3.1.6.2.3

Rewrite the expression.

$5 x = \frac{4 π}{3}$

Step 8.3.2

Divide each term in $5 x = \frac{4 π}{3}$ by $5$ and simplify.

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Step 8.3.2.1

Divide each term in $5 x = \frac{4 π}{3}$ by $5$ .

$\frac{5 x}{5} = \frac{\frac{4 π}{3}}{5}$

Step 8.3.2.2

Simplify the left side.

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Step 8.3.2.2.1

Cancel the common factor of $5$ .

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Step 8.3.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{\frac{4 π}{3}}{5}$

Step 8.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{4 π}{3}}{5}$

Step 8.3.2.3

Simplify the right side.

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Step 8.3.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{4 π}{3} \cdot \frac{1}{5}$

Step 8.3.2.3.2

Multiply $\frac{4 π}{3} \cdot \frac{1}{5}$ .

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Step 8.3.2.3.2.1

Multiply $\frac{4 π}{3}$ by $\frac{1}{5}$ .

$x = \frac{4 π}{3 \cdot 5}$

Step 8.3.2.3.2.2

Multiply $3$ by $5$ .

$x = \frac{4 π}{15}$

Step 9

Find the period of $\sin (5 x - \frac{1}{12} \cdot π)$ .

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Step 9.1

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

$\frac{2 π}{| b |}$

Step 9.2

Replace $b$ with $5$ in the formula for period.

$\frac{2 π}{| 5 |}$

Step 9.3

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$\frac{2 π}{5}$

Step 10

Add $\frac{2 π}{5}$ to every negative angle to get positive angles.

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Step 10.1

Add $\frac{2 π}{5}$ to $- \frac{π}{30}$ to find the positive angle.

$- \frac{π}{30} + \frac{2 π}{5}$

Step 10.2

To write $\frac{2 π}{5}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{2 π}{5} \cdot \frac{6}{6} - \frac{π}{30}$

Step 10.3

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.3.1

Multiply $\frac{2 π}{5}$ by $\frac{6}{6}$ .

$\frac{2 π \cdot 6}{5 \cdot 6} - \frac{π}{30}$

Step 10.3.2

Multiply $5$ by $6$ .

$\frac{2 π \cdot 6}{30} - \frac{π}{30}$

Step 10.4

Combine the numerators over the common denominator.

$\frac{2 π \cdot 6 - π}{30}$

Step 10.5

Simplify the numerator.

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Step 10.5.1

Multiply $6$ by $2$ .

$\frac{12 π - π}{30}$

Step 10.5.2

Subtract $π$ from $12 π$ .

$\frac{11 π}{30}$

Step 10.6

List the new angles.

$x = \frac{11 π}{30}$

Step 11

The period of the $\sin (5 x - \frac{1}{12} \cdot π)$ function is $\frac{2 π}{5}$ so values will repeat every $\frac{2 π}{5}$ radians in both directions.

$x = \frac{4 π}{15} + \frac{2 π n}{5}, \frac{11 π}{30} + \frac{2 π n}{5}$ , for any integer $n$