Precalculus Examples

$104 = 101.8 - 7 \sin (\frac{π}{8} t)$

Step 1

Rewrite the equation as $101.8 - 7 \sin (\frac{π}{8} t) = 104$ .

$101.8 - 7 \sin (\frac{π}{8} t) = 104$

Step 2

Move all terms not containing $\sin (\frac{π}{8} t)$ to the right side of the equation.

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Step 2.1

Subtract $101.8$ from both sides of the equation.

$- 7 \sin (\frac{π}{8} t) = 104 - 101.8$

Step 2.2

Subtract $101.8$ from $104$ .

$- 7 \sin (\frac{π}{8} t) = 2.2$

Step 3

Divide each term in $- 7 \sin (\frac{π}{8} t) = 2.2$ by $- 7$ and simplify.

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Step 3.1

Divide each term in $- 7 \sin (\frac{π}{8} t) = 2.2$ by $- 7$ .

$\frac{- 7 \sin (\frac{π}{8} t)}{- 7} = \frac{2.2}{- 7}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 7$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 7 \sin (\frac{π}{8} t)}{- 7} = \frac{2.2}{- 7}$

Step 3.2.1.2

Divide $\sin (\frac{π}{8} t)$ by $1$ .

$\sin (\frac{π}{8} t) = \frac{2.2}{- 7}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $2.2$ by $- 7$ .

$\sin (\frac{π}{8} t) = - 0.3\overline{142857}$

Step 4

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

$\frac{π}{8} t = \arcsin (- 0.3\overline{142857})$

Step 5

Simplify the left side.

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Step 5.1

Combine $\frac{π}{8}$ and $t$ .

$\frac{π t}{8} = \arcsin (- 0.3\overline{142857})$

Step 6

Simplify the right side.

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Step 6.1

Evaluate $\arcsin (- 0.3\overline{142857})$ .

$\frac{π t}{8} = - 0.31970414$

Step 7

Multiply both sides of the equation by $\frac{8}{π}$ .

$\frac{8}{π} \cdot \frac{π t}{8} = \frac{8}{π} \cdot - 0.31970414$

Step 8

Simplify both sides of the equation.

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Step 8.1

Simplify the left side.

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Step 8.1.1

Simplify $\frac{8}{π} \cdot \frac{π t}{8}$ .

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Step 8.1.1.1

Cancel the common factor of $8$ .

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Step 8.1.1.1.1

Cancel the common factor.

$\frac{8}{π} \cdot \frac{π t}{8} = \frac{8}{π} \cdot - 0.31970414$

Step 8.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π t) = \frac{8}{π} \cdot - 0.31970414$

Step 8.1.1.2

Cancel the common factor of $π$ .

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Step 8.1.1.2.1

Factor $π$ out of $π t$ .

$\frac{1}{π} (π (t)) = \frac{8}{π} \cdot - 0.31970414$

Step 8.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π t) = \frac{8}{π} \cdot - 0.31970414$

Step 8.1.1.2.3

Rewrite the expression.

$t = \frac{8}{π} \cdot - 0.31970414$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $\frac{8}{π} \cdot - 0.31970414$ .

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Step 8.2.1.1

Multiply $\frac{8}{π} \cdot - 0.31970414$ .

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Step 8.2.1.1.1

Combine $\frac{8}{π}$ and $- 0.31970414$ .

$t = \frac{8 \cdot - 0.31970414}{π}$

Step 8.2.1.1.2

Multiply $8$ by $- 0.31970414$ .

$t = \frac{- 2.55763318}{π}$

Step 8.2.1.2

Move the negative in front of the fraction.

$t = - \frac{2.55763318}{π}$

Step 8.2.1.3

Replace $π$ with an approximation.

$t = - \frac{2.55763318}{3.14159265}$

Step 8.2.1.4

Divide $2.55763318$ by $3.14159265$ .

$t = - 1 \cdot 0.81411992$

Step 8.2.1.5

Multiply $- 1$ by $0.81411992$ .

$t = - 0.81411992$

Step 9

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.

$\frac{π t}{8} = 2 (3.14159265) + 0.31970414 + 3.14159265$

Step 10

Simplify the expression to find the second solution.

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Step 10.1

Subtract $2 π$ from $2 (3.14159265) + 0.31970414 + 3.14159265$ .

$\frac{π t}{8} = 2 (3.14159265) + 0.31970414 + 3.14159265 - 2 π$

Step 10.2

The resulting angle of $3.4612968$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.31970414 + 3.14159265$ .

$\frac{π t}{8} = 3.4612968$

Step 10.3

Solve for $t$ .

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Step 10.3.1

Multiply both sides of the equation by $\frac{8}{π}$ .

$\frac{8}{π} \cdot \frac{π t}{8} = \frac{8}{π} \cdot 3.4612968$

Step 10.3.2

Simplify both sides of the equation.

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Step 10.3.2.1

Simplify the left side.

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Step 10.3.2.1.1

Simplify $\frac{8}{π} \cdot \frac{π t}{8}$ .

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Step 10.3.2.1.1.1

Cancel the common factor of $8$ .

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Step 10.3.2.1.1.1.1

Cancel the common factor.

$\frac{8}{π} \cdot \frac{π t}{8} = \frac{8}{π} \cdot 3.4612968$

Step 10.3.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π t) = \frac{8}{π} \cdot 3.4612968$

Step 10.3.2.1.1.2

Cancel the common factor of $π$ .

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Step 10.3.2.1.1.2.1

Factor $π$ out of $π t$ .

$\frac{1}{π} (π (t)) = \frac{8}{π} \cdot 3.4612968$

Step 10.3.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π t) = \frac{8}{π} \cdot 3.4612968$

Step 10.3.2.1.1.2.3

Rewrite the expression.

$t = \frac{8}{π} \cdot 3.4612968$

Step 10.3.2.2

Simplify the right side.

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Step 10.3.2.2.1

Simplify $\frac{8}{π} \cdot 3.4612968$ .

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Step 10.3.2.2.1.1

Multiply $\frac{8}{π} \cdot 3.4612968$ .

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Step 10.3.2.2.1.1.1

Combine $\frac{8}{π}$ and $3.4612968$ .

$t = \frac{8 \cdot 3.4612968}{π}$

Step 10.3.2.2.1.1.2

Multiply $8$ by $3.4612968$ .

$t = \frac{27.69037441}{π}$

Step 10.3.2.2.1.2

Replace $π$ with an approximation.

$t = \frac{27.69037441}{3.14159265}$

Step 10.3.2.2.1.3

Divide $27.69037441$ by $3.14159265$ .

$t = 8.81411992$

Step 11

Find the period of $\sin (\frac{π}{8} t)$ .

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Step 11.1

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

$\frac{2 π}{| b |}$

Step 11.2

Replace $b$ with $\frac{π}{8}$ in the formula for period.

$\frac{2 π}{| \frac{π}{8} |}$

Step 11.3

$\frac{π}{8}$ is approximately $0.39269908$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{8}}$

Step 11.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{8}{π}$

Step 11.5

Cancel the common factor of $π$ .

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Step 11.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{8}{π}$

Step 11.5.2

Cancel the common factor.

$π \cdot 2 \frac{8}{π}$

Step 11.5.3

Rewrite the expression.

$2 \cdot 8$

Step 11.6

Multiply $2$ by $8$ .

$16$

Step 12

Add $16$ to every negative angle to get positive angles.

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Step 12.1

Add $16$ to $- 0.81411992$ to find the positive angle.

$- 0.81411992 + 16$

Step 12.2

Subtract $0.81411992$ from $16$ .

$15.18588007$

Step 12.3

List the new angles.

$t = 15.18588007$

Step 13

The period of the $\sin (\frac{π}{8} t)$ function is $16$ so values will repeat every $16$ radians in both directions.

$t = 8.81411992 + 16 n, 15.18588007 + 16 n$ , for any integer $n$