Trigonometry Examples

$50 = 25 \sin (\frac{2 π}{7} t) + 75$

Step 1

Rewrite the equation as $25 \sin (\frac{2 π}{7} t) + 75 = 50$ .

$25 \sin (\frac{2 π}{7} t) + 75 = 50$

Step 2

Move all terms not containing $\sin (\frac{2 π}{7} t)$ to the right side of the equation.

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Step 2.1

Subtract $75$ from both sides of the equation.

$25 \sin (\frac{2 π}{7} t) = 50 - 75$

Step 2.2

Subtract $75$ from $50$ .

$25 \sin (\frac{2 π}{7} t) = - 25$

Step 3

Divide each term in $25 \sin (\frac{2 π}{7} t) = - 25$ by $25$ and simplify.

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Step 3.1

Divide each term in $25 \sin (\frac{2 π}{7} t) = - 25$ by $25$ .

$\frac{25 \sin (\frac{2 π}{7} t)}{25} = \frac{- 25}{25}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $25$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{25 \sin (\frac{2 π}{7} t)}{25} = \frac{- 25}{25}$

Step 3.2.1.2

Divide $\sin (\frac{2 π}{7} t)$ by $1$ .

$\sin (\frac{2 π}{7} t) = \frac{- 25}{25}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $- 25$ by $25$ .

$\sin (\frac{2 π}{7} t) = - 1$

Step 4

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

$\frac{2 π}{7} t = \arcsin (- 1)$

Step 5

Simplify the left side.

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Step 5.1

Combine $\frac{2 π}{7}$ and $t$ .

$\frac{2 π t}{7} = \arcsin (- 1)$

Step 6

Simplify the right side.

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Step 6.1

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

$\frac{2 π t}{7} = - \frac{π}{2}$

Step 7

Multiply both sides of the equation by $\frac{7}{2 π}$ .

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

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 $\frac{7}{2 π} \cdot \frac{2 π t}{7}$ .

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Step 8.1.1.1

Cancel the common factor of $7$ .

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Step 8.1.1.1.1

Cancel the common factor.

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

Step 8.1.1.1.2

Rewrite the expression.

$\frac{1}{2 π} (2 π t) = \frac{7}{2 π} (- \frac{π}{2})$

Step 8.1.1.2

Cancel the common factor of $2 π$ .

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Step 8.1.1.2.1

Factor $2 π$ out of $2 π t$ .

$\frac{1}{2 π} (2 π (t)) = \frac{7}{2 π} (- \frac{π}{2})$

Step 8.1.1.2.2

Cancel the common factor.

$\frac{1}{2 π} (2 π t) = \frac{7}{2 π} (- \frac{π}{2})$

Step 8.1.1.2.3

Rewrite the expression.

$t = \frac{7}{2 π} (- \frac{π}{2})$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $\frac{7}{2 π} (- \frac{π}{2})$ .

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Step 8.2.1.1

Cancel the common factor of $π$ .

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Step 8.2.1.1.1

Move the leading negative in $- \frac{π}{2}$ into the numerator.

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

Step 8.2.1.1.2

Factor $π$ out of $2 π$ .

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

Step 8.2.1.1.3

Factor $π$ out of $- π$ .

$t = \frac{7}{π \cdot 2} \cdot \frac{π \cdot - 1}{2}$

Step 8.2.1.1.4

Cancel the common factor.

$t = \frac{7}{π \cdot 2} \cdot \frac{π \cdot - 1}{2}$

Step 8.2.1.1.5

Rewrite the expression.

$t = \frac{7}{2} \cdot \frac{- 1}{2}$

Step 8.2.1.2

Multiply $\frac{7}{2}$ by $\frac{- 1}{2}$ .

$t = \frac{7 \cdot - 1}{2 \cdot 2}$

Step 8.2.1.3

Simplify the expression.

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Step 8.2.1.3.1

Multiply $7$ by $- 1$ .

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

Step 8.2.1.3.2

Multiply $2$ by $2$ .

$t = \frac{- 7}{4}$

Step 8.2.1.3.3

Move the negative in front of the fraction.

$t = - \frac{7}{4}$

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.

$\frac{2 π t}{7} = 2 π + \frac{π}{2} + π$

Step 10

Simplify the expression to find the second solution.

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Step 10.1

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

$\frac{2 π t}{7} = 2 π + \frac{π}{2} + π - 2 π$

Step 10.2

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

$\frac{2 π t}{7} = \frac{3 π}{2}$

Step 10.3

Solve for $t$ .

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Step 10.3.1

Multiply both sides of the equation by $\frac{7}{2 π}$ .

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

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 $\frac{7}{2 π} \cdot \frac{2 π t}{7}$ .

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Step 10.3.2.1.1.1

Cancel the common factor of $7$ .

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Step 10.3.2.1.1.1.1

Cancel the common factor.

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

Step 10.3.2.1.1.1.2

Rewrite the expression.

$\frac{1}{2 π} (2 π t) = \frac{7}{2 π} \cdot \frac{3 π}{2}$

Step 10.3.2.1.1.2

Cancel the common factor of $2 π$ .

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Step 10.3.2.1.1.2.1

Factor $2 π$ out of $2 π t$ .

$\frac{1}{2 π} (2 π (t)) = \frac{7}{2 π} \cdot \frac{3 π}{2}$

Step 10.3.2.1.1.2.2

Cancel the common factor.

$\frac{1}{2 π} (2 π t) = \frac{7}{2 π} \cdot \frac{3 π}{2}$

Step 10.3.2.1.1.2.3

Rewrite the expression.

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

Step 10.3.2.2

Simplify the right side.

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Step 10.3.2.2.1

Simplify $\frac{7}{2 π} \cdot \frac{3 π}{2}$ .

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Step 10.3.2.2.1.1

Cancel the common factor of $π$ .

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Step 10.3.2.2.1.1.1

Factor $π$ out of $2 π$ .

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

Step 10.3.2.2.1.1.2

Factor $π$ out of $3 π$ .

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

Step 10.3.2.2.1.1.3

Cancel the common factor.

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

Step 10.3.2.2.1.1.4

Rewrite the expression.

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

Step 10.3.2.2.1.2

Multiply $\frac{7}{2}$ by $\frac{3}{2}$ .

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

Step 10.3.2.2.1.3

Multiply.

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Step 10.3.2.2.1.3.1

Multiply $7$ by $3$ .

$t = \frac{21}{2 \cdot 2}$

Step 10.3.2.2.1.3.2

Multiply $2$ by $2$ .

$t = \frac{21}{4}$

Step 11

Find the period of $\sin (\frac{2 π}{7} t)$ .

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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 $\frac{2 π}{7}$ in the formula for period.

$\frac{2 π}{| \frac{2 π}{7} |}$

Step 11.3

$\frac{2 π}{7}$ is approximately $0.8975979$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2 π}{7}}$

Step 11.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{7}{2 π}$

Step 11.5

Cancel the common factor of $2 π$ .

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Step 11.5.1

Cancel the common factor.

$2 π \frac{7}{2 π}$

Step 11.5.2

Rewrite the expression.

$7$

Step 12

Add $7$ to every negative angle to get positive angles.

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Step 12.1

Add $7$ to $- \frac{7}{4}$ to find the positive angle.

$- \frac{7}{4} + 7$

Step 12.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$7 \cdot \frac{4}{4} - \frac{7}{4}$

Step 12.3

Combine $7$ and $\frac{4}{4}$ .

$\frac{7 \cdot 4}{4} - \frac{7}{4}$

Step 12.4

Combine the numerators over the common denominator.

$\frac{7 \cdot 4 - 7}{4}$

Step 12.5

Simplify the numerator.

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Step 12.5.1

Multiply $7$ by $4$ .

$\frac{28 - 7}{4}$

Step 12.5.2

Subtract $7$ from $28$ .

$\frac{21}{4}$

Step 12.6

List the new angles.

$t = \frac{21}{4}$

Step 13

The period of the $\sin (\frac{2 π}{7} t)$ function is $7$ so values will repeat every $7$ radians in both directions.

$t = \frac{21}{4} + 7 n, \frac{21}{4} + 7 n$ , for any integer $n$

Step 14

Consolidate the answers.

$t = \frac{21}{4} + 7 n$ , for any integer $n$