Precalculus Examples

$\sin (\frac{x}{2} - \frac{π}{4}) = \frac{1}{2}$

Step 1

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

$\frac{x}{2} - \frac{π}{4} = \arcsin (\frac{1}{2})$

Step 2

Simplify the right side.

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Step 2.1

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

$\frac{x}{2} - \frac{π}{4} = \frac{π}{6}$

Step 3

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

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Step 3.1

Add $\frac{π}{4}$ to both sides of the equation.

$\frac{x}{2} = \frac{π}{6} + \frac{π}{4}$

Step 3.2

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

$\frac{x}{2} = \frac{π}{6} \cdot \frac{2}{2} + \frac{π}{4}$

Step 3.3

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

$\frac{x}{2} = \frac{π}{6} \cdot \frac{2}{2} + \frac{π}{4} \cdot \frac{3}{3}$

Step 3.4

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

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Step 3.4.1

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

$\frac{x}{2} = \frac{π \cdot 2}{6 \cdot 2} + \frac{π}{4} \cdot \frac{3}{3}$

Step 3.4.2

Multiply $6$ by $2$ .

$\frac{x}{2} = \frac{π \cdot 2}{12} + \frac{π}{4} \cdot \frac{3}{3}$

Step 3.4.3

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

$\frac{x}{2} = \frac{π \cdot 2}{12} + \frac{π \cdot 3}{4 \cdot 3}$

Step 3.4.4

Multiply $4$ by $3$ .

$\frac{x}{2} = \frac{π \cdot 2}{12} + \frac{π \cdot 3}{12}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{x}{2} = \frac{π \cdot 2 + π \cdot 3}{12}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Move $2$ to the left of $π$ .

$\frac{x}{2} = \frac{2 \cdot π + π \cdot 3}{12}$

Step 3.6.2

Move $3$ to the left of $π$ .

$\frac{x}{2} = \frac{2 π + 3 \cdot π}{12}$

Step 3.6.3

Add $2 π$ and $3 π$ .

$\frac{x}{2} = \frac{5 π}{12}$

Step 4

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{5 π}{12}$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $2$ .

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Step 5.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{5 π}{12}$

Step 5.1.1.2

Rewrite the expression.

$x = 2 \frac{5 π}{12}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Factor $2$ out of $12$ .

$x = 2 \frac{5 π}{2 (6)}$

Step 5.2.1.2

Cancel the common factor.

$x = 2 \frac{5 π}{2 \cdot 6}$

Step 5.2.1.3

Rewrite the expression.

$x = \frac{5 π}{6}$

Step 6

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.

$\frac{x}{2} - \frac{π}{4} = π - \frac{π}{6}$

Step 7

Solve for $x$ .

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Step 7.1

Simplify $π - \frac{π}{6}$ .

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Step 7.1.1

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

$\frac{x}{2} - \frac{π}{4} = π \cdot \frac{6}{6} - \frac{π}{6}$

Step 7.1.2

Combine fractions.

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Step 7.1.2.1

Combine $π$ and $\frac{6}{6}$ .

$\frac{x}{2} - \frac{π}{4} = \frac{π \cdot 6}{6} - \frac{π}{6}$

Step 7.1.2.2

Combine the numerators over the common denominator.

$\frac{x}{2} - \frac{π}{4} = \frac{π \cdot 6 - π}{6}$

Step 7.1.3

Simplify the numerator.

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Step 7.1.3.1

Move $6$ to the left of $π$ .

$\frac{x}{2} - \frac{π}{4} = \frac{6 \cdot π - π}{6}$

Step 7.1.3.2

Subtract $π$ from $6 π$ .

$\frac{x}{2} - \frac{π}{4} = \frac{5 π}{6}$

Step 7.2

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

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Step 7.2.1

Add $\frac{π}{4}$ to both sides of the equation.

$\frac{x}{2} = \frac{5 π}{6} + \frac{π}{4}$

Step 7.2.2

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

$\frac{x}{2} = \frac{5 π}{6} \cdot \frac{2}{2} + \frac{π}{4}$

Step 7.2.3

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

$\frac{x}{2} = \frac{5 π}{6} \cdot \frac{2}{2} + \frac{π}{4} \cdot \frac{3}{3}$

Step 7.2.4

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

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Step 7.2.4.1

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

$\frac{x}{2} = \frac{5 π \cdot 2}{6 \cdot 2} + \frac{π}{4} \cdot \frac{3}{3}$

Step 7.2.4.2

Multiply $6$ by $2$ .

$\frac{x}{2} = \frac{5 π \cdot 2}{12} + \frac{π}{4} \cdot \frac{3}{3}$

Step 7.2.4.3

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

$\frac{x}{2} = \frac{5 π \cdot 2}{12} + \frac{π \cdot 3}{4 \cdot 3}$

Step 7.2.4.4

Multiply $4$ by $3$ .

$\frac{x}{2} = \frac{5 π \cdot 2}{12} + \frac{π \cdot 3}{12}$

Step 7.2.5

Combine the numerators over the common denominator.

$\frac{x}{2} = \frac{5 π \cdot 2 + π \cdot 3}{12}$

Step 7.2.6

Simplify the numerator.

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Step 7.2.6.1

Multiply $2$ by $5$ .

$\frac{x}{2} = \frac{10 π + π \cdot 3}{12}$

Step 7.2.6.2

Move $3$ to the left of $π$ .

$\frac{x}{2} = \frac{10 π + 3 \cdot π}{12}$

Step 7.2.6.3

Add $10 π$ and $3 π$ .

$\frac{x}{2} = \frac{13 π}{12}$

Step 7.3

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{13 π}{12}$

Step 7.4

Simplify both sides of the equation.

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Step 7.4.1

Simplify the left side.

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Step 7.4.1.1

Cancel the common factor of $2$ .

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Step 7.4.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{13 π}{12}$

Step 7.4.1.1.2

Rewrite the expression.

$x = 2 \frac{13 π}{12}$

Step 7.4.2

Simplify the right side.

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Step 7.4.2.1

Cancel the common factor of $2$ .

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Step 7.4.2.1.1

Factor $2$ out of $12$ .

$x = 2 \frac{13 π}{2 (6)}$

Step 7.4.2.1.2

Cancel the common factor.

$x = 2 \frac{13 π}{2 \cdot 6}$

Step 7.4.2.1.3

Rewrite the expression.

$x = \frac{13 π}{6}$

Step 8

Find the period of $\sin (\frac{x}{2} - \frac{π}{4})$ .

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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 $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 8.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 8.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 8.5

Multiply $2$ by $2$ .

$4 π$

Step 9

The period of the $\sin (\frac{x}{2} - \frac{π}{4})$ function is $4 π$ so values will repeat every $4 π$ radians in both directions.

$x = \frac{5 π}{6} + 4 π n, \frac{13 π}{6} + 4 π n$ , for any integer $n$