Precalculus Examples

$30 = 15 \cos (\frac{2 π}{365} t) + 43$

Step 1

Rewrite the equation as $15 \cos (\frac{2 π}{365} t) + 43 = 30$ .

$15 \cos (\frac{2 π}{365} t) + 43 = 30$

Step 2

Move all terms not containing $\cos (\frac{2 π}{365} t)$ to the right side of the equation.

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Step 2.1

Subtract $43$ from both sides of the equation.

$15 \cos (\frac{2 π}{365} t) = 30 - 43$

Step 2.2

Subtract $43$ from $30$ .

$15 \cos (\frac{2 π}{365} t) = - 13$

Step 3

Divide each term in $15 \cos (\frac{2 π}{365} t) = - 13$ by $15$ and simplify.

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Step 3.1

Divide each term in $15 \cos (\frac{2 π}{365} t) = - 13$ by $15$ .

$\frac{15 \cos (\frac{2 π}{365} t)}{15} = \frac{- 13}{15}$

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.

$\frac{15 \cos (\frac{2 π}{365} t)}{15} = \frac{- 13}{15}$

Step 3.2.1.2

Divide $\cos (\frac{2 π}{365} t)$ by $1$ .

$\cos (\frac{2 π}{365} t) = \frac{- 13}{15}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

$\cos (\frac{2 π}{365} t) = - \frac{13}{15}$

Step 4

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

$\frac{2 π}{365} t = \arccos (- \frac{13}{15})$

Step 5

Simplify the left side.

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Step 5.1

Combine $\frac{2 π}{365}$ and $t$ .

$\frac{2 π t}{365} = \arccos (- \frac{13}{15})$

Step 6

Simplify the right side.

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Step 6.1

Evaluate $\arccos (- \frac{13}{15})$ .

$\frac{2 π t}{365} = 2.61927783$

Step 7

Multiply both sides of the equation by $\frac{365}{2 π}$ .

$\frac{365}{2 π} \cdot \frac{2 π t}{365} = \frac{365}{2 π} \cdot 2.61927783$

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 $\frac{365}{2 π} \cdot \frac{2 π t}{365}$ .

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Step 8.1.1.1

Cancel the common factor of $365$ .

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Step 8.1.1.1.1

Cancel the common factor.

$\frac{365}{2 π} \cdot \frac{2 π t}{365} = \frac{365}{2 π} \cdot 2.61927783$

Step 8.1.1.1.2

Rewrite the expression.

$\frac{1}{2 π} (2 π t) = \frac{365}{2 π} \cdot 2.61927783$

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 $2 π t$ .

$\frac{1}{2 π} (2 π (t)) = \frac{365}{2 π} \cdot 2.61927783$

Step 8.1.1.2.2

Cancel the common factor.

$\frac{1}{2 π} (2 π t) = \frac{365}{2 π} \cdot 2.61927783$

Step 8.1.1.2.3

Rewrite the expression.

$t = \frac{365}{2 π} \cdot 2.61927783$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $\frac{365}{2 π} \cdot 2.61927783$ .

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Step 8.2.1.1

Multiply $\frac{365}{2 π} \cdot 2.61927783$ .

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Step 8.2.1.1.1

Combine $\frac{365}{2 π}$ and $2.61927783$ .

$t = \frac{365 \cdot 2.61927783}{2 π}$

Step 8.2.1.1.2

Multiply $365$ by $2.61927783$ .

$t = \frac{956.0364086}{2 π}$

Step 8.2.1.2

Replace $π$ with an approximation.

$t = \frac{956.0364086}{2 \cdot 3.14159265}$

Step 8.2.1.3

Multiply $2$ by $3.14159265$ .

$t = \frac{956.0364086}{6.2831853}$

Step 8.2.1.4

Divide $956.0364086$ by $6.2831853$ .

$t = 152.1579202$

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.

$\frac{2 π t}{365} = 2 (3.14159265) - 2.61927783$

Step 10

Solve for $t$ .

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Step 10.1

Multiply both sides of the equation by $\frac{365}{2 π}$ .

$\frac{365}{2 π} \cdot \frac{2 π t}{365} = \frac{365}{2 π} (2 (3.14159265) - 2.61927783)$

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 $\frac{365}{2 π} \cdot \frac{2 π t}{365}$ .

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Step 10.2.1.1.1

Cancel the common factor of $365$ .

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Step 10.2.1.1.1.1

Cancel the common factor.

$\frac{365}{2 π} \cdot \frac{2 π t}{365} = \frac{365}{2 π} (2 (3.14159265) - 2.61927783)$

Step 10.2.1.1.1.2

Rewrite the expression.

$\frac{1}{2 π} (2 π t) = \frac{365}{2 π} (2 (3.14159265) - 2.61927783)$

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 $2 π t$ .

$\frac{1}{2 π} (2 π (t)) = \frac{365}{2 π} (2 (3.14159265) - 2.61927783)$

Step 10.2.1.1.2.2

Cancel the common factor.

$\frac{1}{2 π} (2 π t) = \frac{365}{2 π} (2 (3.14159265) - 2.61927783)$

Step 10.2.1.1.2.3

Rewrite the expression.

$t = \frac{365}{2 π} (2 (3.14159265) - 2.61927783)$

Step 10.2.2

Simplify the right side.

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Step 10.2.2.1

Simplify $\frac{365}{2 π} (2 (3.14159265) - 2.61927783)$ .

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Step 10.2.2.1.1

Multiply $2$ by $3.14159265$ .

$t = \frac{365}{2 π} (6.2831853 - 2.61927783)$

Step 10.2.2.1.2

Subtract $2.61927783$ from $6.2831853$ .

$t = \frac{365}{2 π} \cdot 3.66390747$

Step 10.2.2.1.3

Multiply $\frac{365}{2 π} \cdot 3.66390747$ .

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Step 10.2.2.1.3.1

Combine $\frac{365}{2 π}$ and $3.66390747$ .

$t = \frac{365 \cdot 3.66390747}{2 π}$

Step 10.2.2.1.3.2

Multiply $365$ by $3.66390747$ .

$t = \frac{1337.32622851}{2 π}$

Step 10.2.2.1.4

Replace $π$ with an approximation.

$t = \frac{1337.32622851}{2 \cdot 3.14159265}$

Step 10.2.2.1.5

Multiply $2$ by $3.14159265$ .

$t = \frac{1337.32622851}{6.2831853}$

Step 10.2.2.1.6

Divide $1337.32622851$ by $6.2831853$ .

$t = 212.84207979$

Step 11

Find the period of $\cos (\frac{2 π}{365} t)$ .

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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 $\frac{2 π}{365}$ in the formula for period.

$\frac{2 π}{| \frac{2 π}{365} |}$

Step 11.3

$\frac{2 π}{365}$ is approximately $0.0172142$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2 π}{365}}$

Step 11.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{365}{2 π}$

Step 11.5

Cancel the common factor of $2 π$ .

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Step 11.5.1

Cancel the common factor.

$2 π \frac{365}{2 π}$

Step 11.5.2

Rewrite the expression.

$365$

Step 12

The period of the $\cos (\frac{2 π}{365} t)$ function is $365$ so values will repeat every $365$ radians in both directions.

$t = 152.1579202 + 365 n, 212.84207979 + 365 n$ , for any integer $n$