Trigonometry Examples

$9 \sin (18 x) + 11 = 2$

Step 1

Move all terms not containing $\sin (18 x)$ to the right side of the equation.

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Step 1.1

Subtract $11$ from both sides of the equation.

$9 \sin (18 x) = 2 - 11$

Step 1.2

Subtract $11$ from $2$ .

$9 \sin (18 x) = - 9$

Step 2

Divide each term in $9 \sin (18 x) = - 9$ by $9$ and simplify.

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Step 2.1

Divide each term in $9 \sin (18 x) = - 9$ by $9$ .

$\frac{9 \sin (18 x)}{9} = \frac{- 9}{9}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $9$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{9 \sin (18 x)}{9} = \frac{- 9}{9}$

Step 2.2.1.2

Divide $\sin (18 x)$ by $1$ .

$\sin (18 x) = \frac{- 9}{9}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $- 9$ by $9$ .

$\sin (18 x) = - 1$

Step 3

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

$18 x = \arcsin (- 1)$

Step 4

Simplify the right side.

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Step 4.1

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

$18 x = - \frac{π}{2}$

Step 5

Divide each term in $18 x = - \frac{π}{2}$ by $18$ and simplify.

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Step 5.1

Divide each term in $18 x = - \frac{π}{2}$ by $18$ .

$\frac{18 x}{18} = \frac{- \frac{π}{2}}{18}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $18$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{18 x}{18} = \frac{- \frac{π}{2}}{18}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{2}}{18}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{2} \cdot \frac{1}{18}$

Step 5.3.2

Multiply $- \frac{π}{2} \cdot \frac{1}{18}$ .

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Step 5.3.2.1

Multiply $\frac{1}{18}$ by $\frac{π}{2}$ .

$x = - \frac{π}{18 \cdot 2}$

Step 5.3.2.2

Multiply $18$ by $2$ .

$x = - \frac{π}{36}$

Step 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.

$18 x = 2 π + \frac{π}{2} + π$

Step 7

Simplify the expression to find the second solution.

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Step 7.1

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

$18 x = 2 π + \frac{π}{2} + π - 2 π$

Step 7.2

The resulting angle of $\frac{3 π}{2}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{2} + π$ .

$18 x = \frac{3 π}{2}$

Step 7.3

Divide each term in $18 x = \frac{3 π}{2}$ by $18$ and simplify.

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Step 7.3.1

Divide each term in $18 x = \frac{3 π}{2}$ by $18$ .

$\frac{18 x}{18} = \frac{\frac{3 π}{2}}{18}$

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of $18$ .

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Step 7.3.2.1.1

Cancel the common factor.

$\frac{18 x}{18} = \frac{\frac{3 π}{2}}{18}$

Step 7.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{18}$

Step 7.3.3

Simplify the right side.

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Step 7.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{18}$

Step 7.3.3.2

Cancel the common factor of $3$ .

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Step 7.3.3.2.1

Factor $3$ out of $3 π$ .

$x = \frac{3 (π)}{2} \cdot \frac{1}{18}$

Step 7.3.3.2.2

Factor $3$ out of $18$ .

$x = \frac{3 (π)}{2} \cdot \frac{1}{3 (6)}$

Step 7.3.3.2.3

Cancel the common factor.

$x = \frac{3 π}{2} \cdot \frac{1}{3 \cdot 6}$

Step 7.3.3.2.4

Rewrite the expression.

$x = \frac{π}{2} \cdot \frac{1}{6}$

Step 7.3.3.3

Multiply $\frac{π}{2}$ by $\frac{1}{6}$ .

$x = \frac{π}{2 \cdot 6}$

Step 7.3.3.4

Multiply $2$ by $6$ .

$x = \frac{π}{12}$

Step 8

Find the period of $\sin (18 x)$ .

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Step 8.1

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

$\frac{2 π}{| b |}$

Step 8.2

Replace $b$ with $18$ in the formula for period.

$\frac{2 π}{| 18 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $18$ is $18$ .

$\frac{2 π}{18}$

Step 8.4

Cancel the common factor of $2$ and $18$ .

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Step 8.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{18}$

Step 8.4.2

Cancel the common factors.

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Step 8.4.2.1

Factor $2$ out of $18$ .

$\frac{2 π}{2 \cdot 9}$

Step 8.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 9}$

Step 8.4.2.3

Rewrite the expression.

$\frac{π}{9}$

Step 9

Add $\frac{π}{9}$ to every negative angle to get positive angles.

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Step 9.1

Add $\frac{π}{9}$ to $- \frac{π}{36}$ to find the positive angle.

$- \frac{π}{36} + \frac{π}{9}$

Step 9.2

To write $\frac{π}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{π}{9} \cdot \frac{4}{4} - \frac{π}{36}$

Step 9.3

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.3.1

Multiply $\frac{π}{9}$ by $\frac{4}{4}$ .

$\frac{π \cdot 4}{9 \cdot 4} - \frac{π}{36}$

Step 9.3.2

Multiply $9$ by $4$ .

$\frac{π \cdot 4}{36} - \frac{π}{36}$

Step 9.4

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{36}$

Step 9.5

Simplify the numerator.

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Step 9.5.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{36}$

Step 9.5.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{36}$

Step 9.6

Cancel the common factor of $3$ and $36$ .

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Step 9.6.1

Factor $3$ out of $3 π$ .

$\frac{3 (π)}{36}$

Step 9.6.2

Cancel the common factors.

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Step 9.6.2.1

Factor $3$ out of $36$ .

$\frac{3 π}{3 \cdot 12}$

Step 9.6.2.2

Cancel the common factor.

$\frac{3 π}{3 \cdot 12}$

Step 9.6.2.3

Rewrite the expression.

$\frac{π}{12}$

Step 9.7

List the new angles.

$x = \frac{π}{12}$

Step 10

The period of the $\sin (18 x)$ function is $\frac{π}{9}$ so values will repeat every $\frac{π}{9}$ radians in both directions.

$x = \frac{π}{12} + \frac{π n}{9}, \frac{π}{12} + \frac{π n}{9}$ , for any integer $n$

Step 11

Consolidate the answers.

$x = \frac{π}{12} + \frac{π n}{9}$ , for any integer $n$