Precalculus Examples

$5 = 7 \cos (\frac{π}{3} t)$

Step 1

Rewrite the equation as $7 \cos (\frac{π}{3} t) = 5$ .

$7 \cos (\frac{π}{3} t) = 5$

Step 2

Divide each term in $7 \cos (\frac{π}{3} t) = 5$ by $7$ and simplify.

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Step 2.1

Divide each term in $7 \cos (\frac{π}{3} t) = 5$ by $7$ .

$\frac{7 \cos (\frac{π}{3} t)}{7} = \frac{5}{7}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $7$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{7 \cos (\frac{π}{3} t)}{7} = \frac{5}{7}$

Step 2.2.1.2

Divide $\cos (\frac{π}{3} t)$ by $1$ .

$\cos (\frac{π}{3} t) = \frac{5}{7}$

Step 3

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

$\frac{π}{3} t = \arccos (\frac{5}{7})$

Step 4

Simplify the left side.

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Step 4.1

Combine $\frac{π}{3}$ and $t$ .

$\frac{π t}{3} = \arccos (\frac{5}{7})$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $\arccos (\frac{5}{7})$ .

$\frac{π t}{3} = 0.77519337$

Step 6

Multiply both sides of the equation by $\frac{3}{π}$ .

$\frac{3}{π} \cdot \frac{π t}{3} = \frac{3}{π} \cdot 0.77519337$

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 $\frac{3}{π} \cdot \frac{π t}{3}$ .

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Step 7.1.1.1

Cancel the common factor of $3$ .

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Step 7.1.1.1.1

Cancel the common factor.

$\frac{3}{π} \cdot \frac{π t}{3} = \frac{3}{π} \cdot 0.77519337$

Step 7.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π t) = \frac{3}{π} \cdot 0.77519337$

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 $π t$ .

$\frac{1}{π} (π (t)) = \frac{3}{π} \cdot 0.77519337$

Step 7.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π t) = \frac{3}{π} \cdot 0.77519337$

Step 7.1.1.2.3

Rewrite the expression.

$t = \frac{3}{π} \cdot 0.77519337$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $\frac{3}{π} \cdot 0.77519337$ .

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Step 7.2.1.1

Multiply $\frac{3}{π} \cdot 0.77519337$ .

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Step 7.2.1.1.1

Combine $\frac{3}{π}$ and $0.77519337$ .

$t = \frac{3 \cdot 0.77519337}{π}$

Step 7.2.1.1.2

Multiply $3$ by $0.77519337$ .

$t = \frac{2.32558011}{π}$

Step 7.2.1.2

Replace $π$ with an approximation.

$t = \frac{2.32558011}{3.14159265}$

Step 7.2.1.3

Divide $2.32558011$ by $3.14159265$ .

$t = 0.74025514$

Step 8

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$\frac{π t}{3} = 2 (3.14159265) - 0.77519337$

Step 9

Solve for $t$ .

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Step 9.1

Multiply both sides of the equation by $\frac{3}{π}$ .

$\frac{3}{π} \cdot \frac{π t}{3} = \frac{3}{π} (2 (3.14159265) - 0.77519337)$

Step 9.2

Simplify both sides of the equation.

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Step 9.2.1

Simplify the left side.

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Step 9.2.1.1

Simplify $\frac{3}{π} \cdot \frac{π t}{3}$ .

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Step 9.2.1.1.1

Cancel the common factor of $3$ .

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Step 9.2.1.1.1.1

Cancel the common factor.

$\frac{3}{π} \cdot \frac{π t}{3} = \frac{3}{π} (2 (3.14159265) - 0.77519337)$

Step 9.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π t) = \frac{3}{π} (2 (3.14159265) - 0.77519337)$

Step 9.2.1.1.2

Cancel the common factor of $π$ .

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Step 9.2.1.1.2.1

Factor $π$ out of $π t$ .

$\frac{1}{π} (π (t)) = \frac{3}{π} (2 (3.14159265) - 0.77519337)$

Step 9.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π t) = \frac{3}{π} (2 (3.14159265) - 0.77519337)$

Step 9.2.1.1.2.3

Rewrite the expression.

$t = \frac{3}{π} (2 (3.14159265) - 0.77519337)$

Step 9.2.2

Simplify the right side.

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Step 9.2.2.1

Simplify $\frac{3}{π} (2 (3.14159265) - 0.77519337)$ .

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Step 9.2.2.1.1

Multiply $2$ by $3.14159265$ .

$t = \frac{3}{π} (6.2831853 - 0.77519337)$

Step 9.2.2.1.2

Subtract $0.77519337$ from $6.2831853$ .

$t = \frac{3}{π} \cdot 5.50799193$

Step 9.2.2.1.3

Multiply $\frac{3}{π} \cdot 5.50799193$ .

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Step 9.2.2.1.3.1

Combine $\frac{3}{π}$ and $5.50799193$ .

$t = \frac{3 \cdot 5.50799193}{π}$

Step 9.2.2.1.3.2

Multiply $3$ by $5.50799193$ .

$t = \frac{16.5239758}{π}$

Step 9.2.2.1.4

Replace $π$ with an approximation.

$t = \frac{16.5239758}{3.14159265}$

Step 9.2.2.1.5

Divide $16.5239758$ by $3.14159265$ .

$t = 5.25974485$

Step 10

Find the period of $\cos (\frac{π}{3} t)$ .

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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 $\frac{π}{3}$ in the formula for period.

$\frac{2 π}{| \frac{π}{3} |}$

Step 10.3

$\frac{π}{3}$ is approximately $1.04719755$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{3}}$

Step 10.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{π}$

Step 10.5

Cancel the common factor of $π$ .

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Step 10.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{3}{π}$

Step 10.5.2

Cancel the common factor.

$π \cdot 2 \frac{3}{π}$

Step 10.5.3

Rewrite the expression.

$2 \cdot 3$

Step 10.6

Multiply $2$ by $3$ .

$6$

Step 11

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

$t = 0.74025514 + 6 n, 5.25974485 + 6 n$ , for any integer $n$