Precalculus Examples

$5.2 = 1.5 \cos (\frac{2 π}{365} t) + 4.3$

Step 1

Rewrite the equation as $1.5 \cos (\frac{2 π}{365} t) + 4.3 = 5.2$ .

$1.5 \cos (\frac{2 π}{365} t) + 4.3 = 5.2$

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 $4.3$ from both sides of the equation.

$1.5 \cos (\frac{2 π}{365} t) = 5.2 - 4.3$

Step 2.2

Subtract $4.3$ from $5.2$ .

$1.5 \cos (\frac{2 π}{365} t) = 0.9$

Step 3

Divide each term in $1.5 \cos (\frac{2 π}{365} t) = 0.9$ by $1.5$ and simplify.

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Step 3.1

Divide each term in $1.5 \cos (\frac{2 π}{365} t) = 0.9$ by $1.5$ .

$\frac{1.5 \cos (\frac{2 π}{365} t)}{1.5} = \frac{0.9}{1.5}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $1.5$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{1.5 \cos (\frac{2 π}{365} t)}{1.5} = \frac{0.9}{1.5}$

Step 3.2.1.2

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

$\cos (\frac{2 π}{365} t) = \frac{0.9}{1.5}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0.9$ by $1.5$ .

$\cos (\frac{2 π}{365} t) = 0.6$

Step 4

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

$\frac{2 π}{365} t = \arccos (0.6)$

Step 5

Simplify the left side.

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Step 5.1

Combine $\frac{2 π}{365}$ and $t$ .

$\frac{2 π t}{365} = \arccos (0.6)$

Step 6

Simplify the right side.

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Step 6.1

Evaluate $\arccos (0.6)$ .

$\frac{2 π t}{365} = 0.92729521$

Step 7

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

$\frac{365}{2 π} \cdot \frac{2 π t}{365} = \frac{365}{2 π} \cdot 0.92729521$

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.

$\frac{365}{2 π} \cdot \frac{2 π t}{365} = \frac{365}{2 π} \cdot 0.92729521$

Step 8.1.1.1.2

Rewrite the expression.

$\frac{1}{2 π} (2 π t) = \frac{365}{2 π} \cdot 0.92729521$

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 $2 π t$ .

$\frac{1}{2 π} (2 π (t)) = \frac{365}{2 π} \cdot 0.92729521$

Step 8.1.1.2.2

Cancel the common factor.

$\frac{1}{2 π} (2 π t) = \frac{365}{2 π} \cdot 0.92729521$

Step 8.1.1.2.3

Rewrite the expression.

$t = \frac{365}{2 π} \cdot 0.92729521$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $\frac{365}{2 π} \cdot 0.92729521$ .

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Step 8.2.1.1

Multiply $\frac{365}{2 π} \cdot 0.92729521$ .

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Step 8.2.1.1.1

Combine $\frac{365}{2 π}$ and $0.92729521$ .

$t = \frac{365 \cdot 0.92729521}{2 π}$

Step 8.2.1.1.2

Multiply $365$ by $0.92729521$ .

$t = \frac{338.46275457}{2 π}$

Step 8.2.1.2

Replace $π$ with an approximation.

$t = \frac{338.46275457}{2 \cdot 3.14159265}$

Step 8.2.1.3

Multiply $2$ by $3.14159265$ .

$t = \frac{338.46275457}{6.2831853}$

Step 8.2.1.4

Divide $338.46275457$ by $6.2831853$ .

$t = 53.86802044$

Step 9

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.

$\frac{2 π t}{365} = 2 (3.14159265) - 0.92729521$

Step 10

Solve for $t$ .

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Step 10.1

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

$\frac{365}{2 π} \cdot \frac{2 π t}{365} = \frac{365}{2 π} (2 (3.14159265) - 0.92729521)$

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.

$\frac{365}{2 π} \cdot \frac{2 π t}{365} = \frac{365}{2 π} (2 (3.14159265) - 0.92729521)$

Step 10.2.1.1.1.2

Rewrite the expression.

$\frac{1}{2 π} (2 π t) = \frac{365}{2 π} (2 (3.14159265) - 0.92729521)$

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 $2 π t$ .

$\frac{1}{2 π} (2 π (t)) = \frac{365}{2 π} (2 (3.14159265) - 0.92729521)$

Step 10.2.1.1.2.2

Cancel the common factor.

$\frac{1}{2 π} (2 π t) = \frac{365}{2 π} (2 (3.14159265) - 0.92729521)$

Step 10.2.1.1.2.3

Rewrite the expression.

$t = \frac{365}{2 π} (2 (3.14159265) - 0.92729521)$

Step 10.2.2

Simplify the right side.

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Step 10.2.2.1

Simplify $\frac{365}{2 π} (2 (3.14159265) - 0.92729521)$ .

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Step 10.2.2.1.1

Multiply $2$ by $3.14159265$ .

$t = \frac{365}{2 π} (6.2831853 - 0.92729521)$

Step 10.2.2.1.2

Subtract $0.92729521$ from $6.2831853$ .

$t = \frac{365}{2 π} \cdot 5.35589008$

Step 10.2.2.1.3

Multiply $\frac{365}{2 π} \cdot 5.35589008$ .

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Step 10.2.2.1.3.1

Combine $\frac{365}{2 π}$ and $5.35589008$ .

$t = \frac{365 \cdot 5.35589008}{2 π}$

Step 10.2.2.1.3.2

Multiply $365$ by $5.35589008$ .

$t = \frac{1954.89988254}{2 π}$

Step 10.2.2.1.4

Replace $π$ with an approximation.

$t = \frac{1954.89988254}{2 \cdot 3.14159265}$

Step 10.2.2.1.5

Multiply $2$ by $3.14159265$ .

$t = \frac{1954.89988254}{6.2831853}$

Step 10.2.2.1.6

Divide $1954.89988254$ by $6.2831853$ .

$t = 311.13197955$

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.

$t = 53.86802044 + 365 n, 311.13197955 + 365 n$ , for any integer $n$