Precalculus Examples

$75 = - 15 \cos (\frac{2 π}{15} t) + 65$

Step 1

Rewrite the equation as $- 15 \cos (\frac{2 π}{15} t) + 65 = 75$ .

$- 15 \cos (\frac{2 π}{15} t) + 65 = 75$

Step 2

Move all terms not containing $\cos (\frac{2 π}{15} t)$ to the right side of the equation.

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Step 2.1

Subtract $65$ from both sides of the equation.

$- 15 \cos (\frac{2 π}{15} t) = 75 - 65$

Step 2.2

Subtract $65$ from $75$ .

$- 15 \cos (\frac{2 π}{15} t) = 10$

Step 3

Divide each term in $- 15 \cos (\frac{2 π}{15} t) = 10$ by $- 15$ and simplify.

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Step 3.1

Divide each term in $- 15 \cos (\frac{2 π}{15} t) = 10$ by $- 15$ .

$\frac{- 15 \cos (\frac{2 π}{15} t)}{- 15} = \frac{10}{- 15}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 15$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 15 \cos (\frac{2 π}{15} t)}{- 15} = \frac{10}{- 15}$

Step 3.2.1.2

Divide $\cos (\frac{2 π}{15} t)$ by $1$ .

$\cos (\frac{2 π}{15} t) = \frac{10}{- 15}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $10$ and $- 15$ .

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Step 3.3.1.1

Factor $5$ out of $10$ .

$\cos (\frac{2 π}{15} t) = \frac{5 (2)}{- 15}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $5$ out of $- 15$ .

$\cos (\frac{2 π}{15} t) = \frac{5 \cdot 2}{5 \cdot - 3}$

Step 3.3.1.2.2

Cancel the common factor.

$\cos (\frac{2 π}{15} t) = \frac{5 \cdot 2}{5 \cdot - 3}$

Step 3.3.1.2.3

Rewrite the expression.

$\cos (\frac{2 π}{15} t) = \frac{2}{- 3}$

Step 3.3.2

Move the negative in front of the fraction.

$\cos (\frac{2 π}{15} t) = - \frac{2}{3}$

Step 4

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

$\frac{2 π}{15} t = \arccos (- \frac{2}{3})$

Step 5

Simplify the left side.

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Step 5.1

Combine $\frac{2 π}{15}$ and $t$ .

$\frac{2 π t}{15} = \arccos (- \frac{2}{3})$

Step 6

Simplify the right side.

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Step 6.1

Evaluate $\arccos (- \frac{2}{3})$ .

$\frac{2 π t}{15} = 2.30052398$

Step 7

Multiply both sides of the equation by $\frac{15}{2 π}$ .

$\frac{15}{2 π} \cdot \frac{2 π t}{15} = \frac{15}{2 π} \cdot 2.30052398$

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 $\frac{15}{2 π} \cdot \frac{2 π t}{15}$ .

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Step 8.1.1.1

Cancel the common factor of $15$ .

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Step 8.1.1.1.1

Cancel the common factor.

$\frac{15}{2 π} \cdot \frac{2 π t}{15} = \frac{15}{2 π} \cdot 2.30052398$

Step 8.1.1.1.2

Rewrite the expression.

$\frac{1}{2 π} (2 π t) = \frac{15}{2 π} \cdot 2.30052398$

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 $2 π t$ .

$\frac{1}{2 π} (2 π (t)) = \frac{15}{2 π} \cdot 2.30052398$

Step 8.1.1.2.2

Cancel the common factor.

$\frac{1}{2 π} (2 π t) = \frac{15}{2 π} \cdot 2.30052398$

Step 8.1.1.2.3

Rewrite the expression.

$t = \frac{15}{2 π} \cdot 2.30052398$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $\frac{15}{2 π} \cdot 2.30052398$ .

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Step 8.2.1.1

Multiply $\frac{15}{2 π} \cdot 2.30052398$ .

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Step 8.2.1.1.1

Combine $\frac{15}{2 π}$ and $2.30052398$ .

$t = \frac{15 \cdot 2.30052398}{2 π}$

Step 8.2.1.1.2

Multiply $15$ by $2.30052398$ .

$t = \frac{34.50785974}{2 π}$

Step 8.2.1.2

Replace $π$ with an approximation.

$t = \frac{34.50785974}{2 \cdot 3.14159265}$

Step 8.2.1.3

Multiply $2$ by $3.14159265$ .

$t = \frac{34.50785974}{6.2831853}$

Step 8.2.1.4

Divide $34.50785974$ by $6.2831853$ .

$t = 5.49209645$

Step 9

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

$\frac{2 π t}{15} = 2 (3.14159265) - 2.30052398$

Step 10

Solve for $t$ .

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Step 10.1

Multiply both sides of the equation by $\frac{15}{2 π}$ .

$\frac{15}{2 π} \cdot \frac{2 π t}{15} = \frac{15}{2 π} (2 (3.14159265) - 2.30052398)$

Step 10.2

Simplify both sides of the equation.

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Step 10.2.1

Simplify the left side.

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Step 10.2.1.1

Simplify $\frac{15}{2 π} \cdot \frac{2 π t}{15}$ .

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Step 10.2.1.1.1

Cancel the common factor of $15$ .

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Step 10.2.1.1.1.1

Cancel the common factor.

$\frac{15}{2 π} \cdot \frac{2 π t}{15} = \frac{15}{2 π} (2 (3.14159265) - 2.30052398)$

Step 10.2.1.1.1.2

Rewrite the expression.

$\frac{1}{2 π} (2 π t) = \frac{15}{2 π} (2 (3.14159265) - 2.30052398)$

Step 10.2.1.1.2

Cancel the common factor of $2 π$ .

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Step 10.2.1.1.2.1

Factor $2 π$ out of $2 π t$ .

$\frac{1}{2 π} (2 π (t)) = \frac{15}{2 π} (2 (3.14159265) - 2.30052398)$

Step 10.2.1.1.2.2

Cancel the common factor.

$\frac{1}{2 π} (2 π t) = \frac{15}{2 π} (2 (3.14159265) - 2.30052398)$

Step 10.2.1.1.2.3

Rewrite the expression.

$t = \frac{15}{2 π} (2 (3.14159265) - 2.30052398)$

Step 10.2.2

Simplify the right side.

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Step 10.2.2.1

Simplify $\frac{15}{2 π} (2 (3.14159265) - 2.30052398)$ .

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Step 10.2.2.1.1

Multiply $2$ by $3.14159265$ .

$t = \frac{15}{2 π} (6.2831853 - 2.30052398)$

Step 10.2.2.1.2

Subtract $2.30052398$ from $6.2831853$ .

$t = \frac{15}{2 π} \cdot 3.98266132$

Step 10.2.2.1.3

Multiply $\frac{15}{2 π} \cdot 3.98266132$ .

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Step 10.2.2.1.3.1

Combine $\frac{15}{2 π}$ and $3.98266132$ .

$t = \frac{15 \cdot 3.98266132}{2 π}$

Step 10.2.2.1.3.2

Multiply $15$ by $3.98266132$ .

$t = \frac{59.73991986}{2 π}$

Step 10.2.2.1.4

Replace $π$ with an approximation.

$t = \frac{59.73991986}{2 \cdot 3.14159265}$

Step 10.2.2.1.5

Multiply $2$ by $3.14159265$ .

$t = \frac{59.73991986}{6.2831853}$

Step 10.2.2.1.6

Divide $59.73991986$ by $6.2831853$ .

$t = 9.50790354$

Step 11

Find the period of $\cos (\frac{2 π}{15} t)$ .

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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 $\frac{2 π}{15}$ in the formula for period.

$\frac{2 π}{| \frac{2 π}{15} |}$

Step 11.3

$\frac{2 π}{15}$ is approximately $0.41887902$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2 π}{15}}$

Step 11.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{15}{2 π}$

Step 11.5

Cancel the common factor of $2 π$ .

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Step 11.5.1

Cancel the common factor.

$2 π \frac{15}{2 π}$

Step 11.5.2

Rewrite the expression.

$15$

Step 12

The period of the $\cos (\frac{2 π}{15} t)$ function is $15$ so values will repeat every $15$ radians in both directions.

$t = 5.49209645 + 15 n, 9.50790354 + 15 n$ , for any integer $n$