Trigonometry Examples

$\sin (\frac{11}{20} x + \frac{π}{12}) = - \frac{1}{2}$

Step 1

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

$\frac{11}{20} x + \frac{π}{12} = \arcsin (- \frac{1}{2})$

Step 2

Simplify the left side.

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Step 2.1

Combine $\frac{11}{20}$ and $x$ .

$\frac{11 x}{20} + \frac{π}{12} = \arcsin (- \frac{1}{2})$

Step 3

Simplify the right side.

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Step 3.1

The exact value of $\arcsin (- \frac{1}{2})$ is $- \frac{π}{6}$ .

$\frac{11 x}{20} + \frac{π}{12} = - \frac{π}{6}$

Step 4

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

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Step 4.1

Subtract $\frac{π}{12}$ from both sides of the equation.

$\frac{11 x}{20} = - \frac{π}{6} - \frac{π}{12}$

Step 4.2

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

$\frac{11 x}{20} = - \frac{π}{6} \cdot \frac{2}{2} - \frac{π}{12}$

Step 4.3

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

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Step 4.3.1

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

$\frac{11 x}{20} = - \frac{π \cdot 2}{6 \cdot 2} - \frac{π}{12}$

Step 4.3.2

Multiply $6$ by $2$ .

$\frac{11 x}{20} = - \frac{π \cdot 2}{12} - \frac{π}{12}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{11 x}{20} = \frac{- π \cdot 2 - π}{12}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $2$ by $- 1$ .

$\frac{11 x}{20} = \frac{- 2 π - π}{12}$

Step 4.5.2

Subtract $π$ from $- 2 π$ .

$\frac{11 x}{20} = \frac{- 3 π}{12}$

Step 4.6

Cancel the common factor of $- 3$ and $12$ .

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Step 4.6.1

Factor $3$ out of $- 3 π$ .

$\frac{11 x}{20} = \frac{3 (- π)}{12}$

Step 4.6.2

Cancel the common factors.

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Step 4.6.2.1

Factor $3$ out of $12$ .

$\frac{11 x}{20} = \frac{3 (- π)}{3 (4)}$

Step 4.6.2.2

Cancel the common factor.

$\frac{11 x}{20} = \frac{3 (- π)}{3 \cdot 4}$

Step 4.6.2.3

Rewrite the expression.

$\frac{11 x}{20} = \frac{- π}{4}$

Step 4.7

Move the negative in front of the fraction.

$\frac{11 x}{20} = - \frac{π}{4}$

Step 5

Multiply both sides of the equation by $\frac{20}{11}$ .

$\frac{20}{11} \cdot \frac{11 x}{20} = \frac{20}{11} (- \frac{π}{4})$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $\frac{20}{11} \cdot \frac{11 x}{20}$ .

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Step 6.1.1.1

Cancel the common factor of $20$ .

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Step 6.1.1.1.1

Cancel the common factor.

$\frac{20}{11} \cdot \frac{11 x}{20} = \frac{20}{11} (- \frac{π}{4})$

Step 6.1.1.1.2

Rewrite the expression.

$\frac{1}{11} (11 x) = \frac{20}{11} (- \frac{π}{4})$

Step 6.1.1.2

Cancel the common factor of $11$ .

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Step 6.1.1.2.1

Factor $11$ out of $11 x$ .

$\frac{1}{11} (11 (x)) = \frac{20}{11} (- \frac{π}{4})$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{1}{11} (11 x) = \frac{20}{11} (- \frac{π}{4})$

Step 6.1.1.2.3

Rewrite the expression.

$x = \frac{20}{11} (- \frac{π}{4})$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{20}{11} (- \frac{π}{4})$ .

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Step 6.2.1.1

Cancel the common factor of $4$ .

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Step 6.2.1.1.1

Move the leading negative in $- \frac{π}{4}$ into the numerator.

$x = \frac{20}{11} \cdot \frac{- π}{4}$

Step 6.2.1.1.2

Factor $4$ out of $20$ .

$x = \frac{4 (5)}{11} \cdot \frac{- π}{4}$

Step 6.2.1.1.3

Cancel the common factor.

$x = \frac{4 \cdot 5}{11} \cdot \frac{- π}{4}$

Step 6.2.1.1.4

Rewrite the expression.

$x = \frac{5}{11} (- π)$

Step 6.2.1.2

Combine $\frac{5}{11}$ and $π$ .

$x = - \frac{5 π}{11}$

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.

$\frac{11 x}{20} + \frac{π}{12} = 2 π + \frac{π}{6} + π$

Step 8

Simplify the expression to find the second solution.

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Step 8.1

Subtract $2 π$ from $2 π + \frac{π}{6} + π$ .

$\frac{11 x}{20} + \frac{π}{12} = 2 π + \frac{π}{6} + π - 2 π$

Step 8.2

The resulting angle of $\frac{7 π}{6}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{6} + π$ .

$\frac{11 x}{20} + \frac{π}{12} = \frac{7 π}{6}$

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

Subtract $\frac{π}{12}$ from both sides of the equation.

$\frac{11 x}{20} = \frac{7 π}{6} - \frac{π}{12}$

Step 8.3.1.2

To write $\frac{7 π}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{11 x}{20} = \frac{7 π}{6} \cdot \frac{2}{2} - \frac{π}{12}$

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 $\frac{7 π}{6}$ by $\frac{2}{2}$ .

$\frac{11 x}{20} = \frac{7 π \cdot 2}{6 \cdot 2} - \frac{π}{12}$

Step 8.3.1.3.2

Multiply $6$ by $2$ .

$\frac{11 x}{20} = \frac{7 π \cdot 2}{12} - \frac{π}{12}$

Step 8.3.1.4

Combine the numerators over the common denominator.

$\frac{11 x}{20} = \frac{7 π \cdot 2 - π}{12}$

Step 8.3.1.5

Simplify the numerator.

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Step 8.3.1.5.1

Multiply $2$ by $7$ .

$\frac{11 x}{20} = \frac{14 π - π}{12}$

Step 8.3.1.5.2

Subtract $π$ from $14 π$ .

$\frac{11 x}{20} = \frac{13 π}{12}$

Step 8.3.2

Multiply both sides of the equation by $\frac{20}{11}$ .

$\frac{20}{11} \cdot \frac{11 x}{20} = \frac{20}{11} \cdot \frac{13 π}{12}$

Step 8.3.3

Simplify both sides of the equation.

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Step 8.3.3.1

Simplify the left side.

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Step 8.3.3.1.1

Simplify $\frac{20}{11} \cdot \frac{11 x}{20}$ .

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Step 8.3.3.1.1.1

Cancel the common factor of $20$ .

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Step 8.3.3.1.1.1.1

Cancel the common factor.

$\frac{20}{11} \cdot \frac{11 x}{20} = \frac{20}{11} \cdot \frac{13 π}{12}$

Step 8.3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{11} (11 x) = \frac{20}{11} \cdot \frac{13 π}{12}$

Step 8.3.3.1.1.2

Cancel the common factor of $11$ .

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Step 8.3.3.1.1.2.1

Factor $11$ out of $11 x$ .

$\frac{1}{11} (11 (x)) = \frac{20}{11} \cdot \frac{13 π}{12}$

Step 8.3.3.1.1.2.2

Cancel the common factor.

$\frac{1}{11} (11 x) = \frac{20}{11} \cdot \frac{13 π}{12}$

Step 8.3.3.1.1.2.3

Rewrite the expression.

$x = \frac{20}{11} \cdot \frac{13 π}{12}$

Step 8.3.3.2

Simplify the right side.

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Step 8.3.3.2.1

Simplify $\frac{20}{11} \cdot \frac{13 π}{12}$ .

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Step 8.3.3.2.1.1

Cancel the common factor of $4$ .

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Step 8.3.3.2.1.1.1

Factor $4$ out of $20$ .

$x = \frac{4 (5)}{11} \cdot \frac{13 π}{12}$

Step 8.3.3.2.1.1.2

Factor $4$ out of $12$ .

$x = \frac{4 \cdot 5}{11} \cdot \frac{13 π}{4 \cdot 3}$

Step 8.3.3.2.1.1.3

Cancel the common factor.

$x = \frac{4 \cdot 5}{11} \cdot \frac{13 π}{4 \cdot 3}$

Step 8.3.3.2.1.1.4

Rewrite the expression.

$x = \frac{5}{11} \cdot \frac{13 π}{3}$

Step 8.3.3.2.1.2

Multiply $\frac{5}{11}$ by $\frac{13 π}{3}$ .

$x = \frac{5 (13 π)}{11 \cdot 3}$

Step 8.3.3.2.1.3

Multiply.

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Step 8.3.3.2.1.3.1

Multiply $13$ by $5$ .

$x = \frac{65 π}{11 \cdot 3}$

Step 8.3.3.2.1.3.2

Multiply $11$ by $3$ .

$x = \frac{65 π}{33}$

Step 9

Find the period of $\sin (\frac{11}{20} x + \frac{π}{12})$ .

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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 $\frac{11}{20}$ in the formula for period.

$\frac{2 π}{| \frac{11}{20} |}$

Step 9.3

$\frac{11}{20}$ is approximately $0.55$ which is positive so remove the absolute value

$\frac{2 π}{\frac{11}{20}}$

Step 9.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{20}{11}$

Step 9.5

Multiply $2 π \frac{20}{11}$ .

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Step 9.5.1

Combine $\frac{20}{11}$ and $2$ .

$\frac{20 \cdot 2}{11} π$

Step 9.5.2

Multiply $20$ by $2$ .

$\frac{40}{11} π$

Step 9.5.3

Combine $\frac{40}{11}$ and $π$ .

$\frac{40 π}{11}$

Step 10

Add $\frac{40 π}{11}$ to every negative angle to get positive angles.

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Step 10.1

Add $\frac{40 π}{11}$ to $- \frac{5 π}{11}$ to find the positive angle.

$- \frac{5 π}{11} + \frac{40 π}{11}$

Step 10.2

Combine the numerators over the common denominator.

$\frac{40 π - 5 π}{11}$

Step 10.3

Subtract $5 π$ from $40 π$ .

$\frac{35 π}{11}$

Step 10.4

List the new angles.

$x = \frac{35 π}{11}$

Step 11

The period of the $\sin (\frac{11}{20} x + \frac{π}{12})$ function is $\frac{40 π}{11}$ so values will repeat every $\frac{40 π}{11}$ radians in both directions.

$x = \frac{65 π}{33} + \frac{40 π n}{11}, \frac{35 π}{11} + \frac{40 π n}{11}$ , for any integer $n$