Precalculus Examples

$\sin (\frac{x}{3}) = \frac{1}{2}$

Step 1

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

$\frac{x}{3} = \arcsin (\frac{1}{2})$

Step 2

Simplify the right side.

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Step 2.1

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

$\frac{x}{3} = \frac{π}{6}$

Step 3

Multiply both sides of the equation by $3$ .

$3 \frac{x}{3} = 3 \frac{π}{6}$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $3$ .

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Step 4.1.1.1

Cancel the common factor.

$3 \frac{x}{3} = 3 \frac{π}{6}$

Step 4.1.1.2

Rewrite the expression.

$x = 3 \frac{π}{6}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Factor $3$ out of $6$ .

$x = 3 \frac{π}{3 (2)}$

Step 4.2.1.2

Cancel the common factor.

$x = 3 \frac{π}{3 \cdot 2}$

Step 4.2.1.3

Rewrite the expression.

$x = \frac{π}{2}$

Step 5

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$\frac{x}{3} = π - \frac{π}{6}$

Step 6

Solve for $x$ .

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Step 6.1

Multiply both sides of the equation by $3$ .

$3 \frac{x}{3} = 3 (π - \frac{π}{6})$

Step 6.2

Simplify both sides of the equation.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Cancel the common factor of $3$ .

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Step 6.2.1.1.1

Cancel the common factor.

$3 \frac{x}{3} = 3 (π - \frac{π}{6})$

Step 6.2.1.1.2

Rewrite the expression.

$x = 3 (π - \frac{π}{6})$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $3 (π - \frac{π}{6})$ .

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Step 6.2.2.1.1

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

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

Step 6.2.2.1.2

Simplify terms.

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Step 6.2.2.1.2.1

Combine $π$ and $\frac{6}{6}$ .

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

Step 6.2.2.1.2.2

Combine the numerators over the common denominator.

$x = 3 \frac{π \cdot 6 - π}{6}$

Step 6.2.2.1.2.3

Cancel the common factor of $3$ .

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Step 6.2.2.1.2.3.1

Factor $3$ out of $6$ .

$x = 3 \frac{π \cdot 6 - π}{3 (2)}$

Step 6.2.2.1.2.3.2

Cancel the common factor.

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

Step 6.2.2.1.2.3.3

Rewrite the expression.

$x = \frac{π \cdot 6 - π}{2}$

Step 6.2.2.1.3

Simplify the numerator.

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Step 6.2.2.1.3.1

Move $6$ to the left of $π$ .

$x = \frac{6 \cdot π - π}{2}$

Step 6.2.2.1.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{2}$

Step 7

Find the period of $\sin (\frac{x}{3})$ .

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

Replace $b$ with $\frac{1}{3}$ in the formula for period.

$\frac{2 π}{| \frac{1}{3} |}$

Step 7.3

$\frac{1}{3}$ is approximately $0.\overline{3}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{3}}$

Step 7.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 3$

Step 7.5

Multiply $3$ by $2$ .

$6 π$

Step 8

The period of the $\sin (\frac{x}{3})$ function is $6 π$ so values will repeat every $6 π$ radians in both directions.

$x = \frac{π}{2} + 6 π n, \frac{5 π}{2} + 6 π n$ , for any integer $n$