Precalculus Examples

$10 \sqrt{3} = 20 \sin (\frac{π t}{4} - \frac{π}{2})$

Step 1

Rewrite the equation as $20 \sin (\frac{π t}{4} - \frac{π}{2}) = 10 \sqrt{3}$ .

$20 \sin (\frac{π t}{4} - \frac{π}{2}) = 10 \sqrt{3}$

Step 2

Divide each term in $20 \sin (\frac{π t}{4} - \frac{π}{2}) = 10 \sqrt{3}$ by $20$ and simplify.

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Step 2.1

Divide each term in $20 \sin (\frac{π t}{4} - \frac{π}{2}) = 10 \sqrt{3}$ by $20$ .

$\frac{20 \sin (\frac{π t}{4} - \frac{π}{2})}{20} = \frac{10 \sqrt{3}}{20}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $20$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{20 \sin (\frac{π t}{4} - \frac{π}{2})}{20} = \frac{10 \sqrt{3}}{20}$

Step 2.2.1.2

Divide $\sin (\frac{π t}{4} - \frac{π}{2})$ by $1$ .

$\sin (\frac{π t}{4} - \frac{π}{2}) = \frac{10 \sqrt{3}}{20}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $10$ and $20$ .

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Step 2.3.1.1

Factor $10$ out of $10 \sqrt{3}$ .

$\sin (\frac{π t}{4} - \frac{π}{2}) = \frac{10 (\sqrt{3})}{20}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $10$ out of $20$ .

$\sin (\frac{π t}{4} - \frac{π}{2}) = \frac{10 \sqrt{3}}{10 \cdot 2}$

Step 2.3.1.2.2

Cancel the common factor.

$\sin (\frac{π t}{4} - \frac{π}{2}) = \frac{10 \sqrt{3}}{10 \cdot 2}$

Step 2.3.1.2.3

Rewrite the expression.

$\sin (\frac{π t}{4} - \frac{π}{2}) = \frac{\sqrt{3}}{2}$

Step 3

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

$\frac{π t}{4} - \frac{π}{2} = \arcsin (\frac{\sqrt{3}}{2})$

Step 4

Simplify the right side.

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Step 4.1

The exact value of $\arcsin (\frac{\sqrt{3}}{2})$ is $\frac{π}{3}$ .

$\frac{π t}{4} - \frac{π}{2} = \frac{π}{3}$

Step 5

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

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Step 5.1

Add $\frac{π}{2}$ to both sides of the equation.

$\frac{π t}{4} = \frac{π}{3} + \frac{π}{2}$

Step 5.2

To write $\frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.3

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 5.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.1

Multiply $\frac{π}{3}$ by $\frac{2}{2}$ .

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

Step 5.4.2

Multiply $3$ by $2$ .

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

Step 5.4.3

Multiply $\frac{π}{2}$ by $\frac{3}{3}$ .

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

Step 5.4.4

Multiply $2$ by $3$ .

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

Step 5.5

Combine the numerators over the common denominator.

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

Step 5.6

Simplify the numerator.

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Step 5.6.1

Move $2$ to the left of $π$ .

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

Step 5.6.2

Move $3$ to the left of $π$ .

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

Step 5.6.3

Add $2 π$ and $3 π$ .

$\frac{π t}{4} = \frac{5 π}{6}$

Step 6

Multiply both sides of the equation by $\frac{4}{π}$ .

$\frac{4}{π} \cdot \frac{π t}{4} = \frac{4}{π} \cdot \frac{5 π}{6}$

Step 7

Simplify both sides of the equation.

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Step 7.1

Simplify the left side.

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Step 7.1.1

Simplify $\frac{4}{π} \cdot \frac{π t}{4}$ .

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Step 7.1.1.1

Cancel the common factor of $4$ .

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Step 7.1.1.1.1

Cancel the common factor.

$\frac{4}{π} \cdot \frac{π t}{4} = \frac{4}{π} \cdot \frac{5 π}{6}$

Step 7.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π t) = \frac{4}{π} \cdot \frac{5 π}{6}$

Step 7.1.1.2

Cancel the common factor of $π$ .

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Step 7.1.1.2.1

Factor $π$ out of $π t$ .

$\frac{1}{π} (π (t)) = \frac{4}{π} \cdot \frac{5 π}{6}$

Step 7.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π t) = \frac{4}{π} \cdot \frac{5 π}{6}$

Step 7.1.1.2.3

Rewrite the expression.

$t = \frac{4}{π} \cdot \frac{5 π}{6}$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $\frac{4}{π} \cdot \frac{5 π}{6}$ .

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Step 7.2.1.1

Cancel the common factor of $2$ .

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Step 7.2.1.1.1

Factor $2$ out of $4$ .

$t = \frac{2 (2)}{π} \cdot \frac{5 π}{6}$

Step 7.2.1.1.2

Factor $2$ out of $6$ .

$t = \frac{2 \cdot 2}{π} \cdot \frac{5 π}{2 \cdot 3}$

Step 7.2.1.1.3

Cancel the common factor.

$t = \frac{2 \cdot 2}{π} \cdot \frac{5 π}{2 \cdot 3}$

Step 7.2.1.1.4

Rewrite the expression.

$t = \frac{2}{π} \cdot \frac{5 π}{3}$

Step 7.2.1.2

Cancel the common factor of $π$ .

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Step 7.2.1.2.1

Factor $π$ out of $5 π$ .

$t = \frac{2}{π} \cdot \frac{π \cdot 5}{3}$

Step 7.2.1.2.2

Cancel the common factor.

$t = \frac{2}{π} \cdot \frac{π \cdot 5}{3}$

Step 7.2.1.2.3

Rewrite the expression.

$t = 2 (\frac{5}{3})$

Step 7.2.1.3

Combine $2$ and $\frac{5}{3}$ .

$t = \frac{2 \cdot 5}{3}$

Step 7.2.1.4

Multiply $2$ by $5$ .

$t = \frac{10}{3}$

Step 8

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.

$\frac{π t}{4} - \frac{π}{2} = π - \frac{π}{3}$

Step 9

Solve for $t$ .

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Step 9.1

Simplify $π - \frac{π}{3}$ .

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Step 9.1.1

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

$\frac{π t}{4} - \frac{π}{2} = π \cdot \frac{3}{3} - \frac{π}{3}$

Step 9.1.2

Combine fractions.

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Step 9.1.2.1

Combine $π$ and $\frac{3}{3}$ .

$\frac{π t}{4} - \frac{π}{2} = \frac{π \cdot 3}{3} - \frac{π}{3}$

Step 9.1.2.2

Combine the numerators over the common denominator.

$\frac{π t}{4} - \frac{π}{2} = \frac{π \cdot 3 - π}{3}$

Step 9.1.3

Simplify the numerator.

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Step 9.1.3.1

Move $3$ to the left of $π$ .

$\frac{π t}{4} - \frac{π}{2} = \frac{3 \cdot π - π}{3}$

Step 9.1.3.2

Subtract $π$ from $3 π$ .

$\frac{π t}{4} - \frac{π}{2} = \frac{2 π}{3}$

Step 9.2

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

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Step 9.2.1

Add $\frac{π}{2}$ to both sides of the equation.

$\frac{π t}{4} = \frac{2 π}{3} + \frac{π}{2}$

Step 9.2.2

To write $\frac{2 π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 9.2.3

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 9.2.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.2.4.1

Multiply $\frac{2 π}{3}$ by $\frac{2}{2}$ .

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

Step 9.2.4.2

Multiply $3$ by $2$ .

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

Step 9.2.4.3

Multiply $\frac{π}{2}$ by $\frac{3}{3}$ .

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

Step 9.2.4.4

Multiply $2$ by $3$ .

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

Step 9.2.5

Combine the numerators over the common denominator.

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

Step 9.2.6

Simplify the numerator.

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Step 9.2.6.1

Multiply $2$ by $2$ .

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

Step 9.2.6.2

Move $3$ to the left of $π$ .

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

Step 9.2.6.3

Add $4 π$ and $3 π$ .

$\frac{π t}{4} = \frac{7 π}{6}$

Step 9.3

Multiply both sides of the equation by $\frac{4}{π}$ .

$\frac{4}{π} \cdot \frac{π t}{4} = \frac{4}{π} \cdot \frac{7 π}{6}$

Step 9.4

Simplify both sides of the equation.

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Step 9.4.1

Simplify the left side.

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Step 9.4.1.1

Simplify $\frac{4}{π} \cdot \frac{π t}{4}$ .

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Step 9.4.1.1.1

Cancel the common factor of $4$ .

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Step 9.4.1.1.1.1

Cancel the common factor.

$\frac{4}{π} \cdot \frac{π t}{4} = \frac{4}{π} \cdot \frac{7 π}{6}$

Step 9.4.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π t) = \frac{4}{π} \cdot \frac{7 π}{6}$

Step 9.4.1.1.2

Cancel the common factor of $π$ .

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Step 9.4.1.1.2.1

Factor $π$ out of $π t$ .

$\frac{1}{π} (π (t)) = \frac{4}{π} \cdot \frac{7 π}{6}$

Step 9.4.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π t) = \frac{4}{π} \cdot \frac{7 π}{6}$

Step 9.4.1.1.2.3

Rewrite the expression.

$t = \frac{4}{π} \cdot \frac{7 π}{6}$

Step 9.4.2

Simplify the right side.

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Step 9.4.2.1

Simplify $\frac{4}{π} \cdot \frac{7 π}{6}$ .

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Step 9.4.2.1.1

Cancel the common factor of $2$ .

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Step 9.4.2.1.1.1

Factor $2$ out of $4$ .

$t = \frac{2 (2)}{π} \cdot \frac{7 π}{6}$

Step 9.4.2.1.1.2

Factor $2$ out of $6$ .

$t = \frac{2 \cdot 2}{π} \cdot \frac{7 π}{2 \cdot 3}$

Step 9.4.2.1.1.3

Cancel the common factor.

$t = \frac{2 \cdot 2}{π} \cdot \frac{7 π}{2 \cdot 3}$

Step 9.4.2.1.1.4

Rewrite the expression.

$t = \frac{2}{π} \cdot \frac{7 π}{3}$

Step 9.4.2.1.2

Cancel the common factor of $π$ .

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Step 9.4.2.1.2.1

Factor $π$ out of $7 π$ .

$t = \frac{2}{π} \cdot \frac{π \cdot 7}{3}$

Step 9.4.2.1.2.2

Cancel the common factor.

$t = \frac{2}{π} \cdot \frac{π \cdot 7}{3}$

Step 9.4.2.1.2.3

Rewrite the expression.

$t = 2 (\frac{7}{3})$

Step 9.4.2.1.3

Combine $2$ and $\frac{7}{3}$ .

$t = \frac{2 \cdot 7}{3}$

Step 9.4.2.1.4

Multiply $2$ by $7$ .

$t = \frac{14}{3}$

Step 10

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

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Step 10.1

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

$\frac{2 π}{| b |}$

Step 10.2

Replace $b$ with $\frac{π}{4}$ in the formula for period.

$\frac{2 π}{| \frac{π}{4} |}$

Step 10.3

$\frac{π}{4}$ is approximately $0.78539816$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{4}}$

Step 10.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{π}$

Step 10.5

Cancel the common factor of $π$ .

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Step 10.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{4}{π}$

Step 10.5.2

Cancel the common factor.

$π \cdot 2 \frac{4}{π}$

Step 10.5.3

Rewrite the expression.

$2 \cdot 4$

Step 10.6

Multiply $2$ by $4$ .

$8$

Step 11

The period of the $\sin (\frac{π t}{4} - \frac{π}{2})$ function is $8$ so values will repeat every $8$ radians in both directions.

$t = \frac{10}{3} + 8 n, \frac{14}{3} + 8 n$ , for any integer $n$