Precalculus Examples

$380 = - 21 \cos (\frac{2 π}{29.5} t) + 384$

Step 1

Rewrite the equation as $- 21 \cos (\frac{2 π}{29.5} t) + 384 = 380$ .

$- 21 \cos (\frac{2 π}{29.5} t) + 384 = 380$

Step 2

Move all terms not containing $\cos (\frac{2 π}{29.5} t)$ to the right side of the equation.

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Step 2.1

Subtract $384$ from both sides of the equation.

$- 21 \cos (\frac{2 π}{29.5} t) = 380 - 384$

Step 2.2

Subtract $384$ from $380$ .

$- 21 \cos (\frac{2 π}{29.5} t) = - 4$

Step 3

Divide each term in $- 21 \cos (\frac{2 π}{29.5} t) = - 4$ by $- 21$ and simplify.

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Step 3.1

Divide each term in $- 21 \cos (\frac{2 π}{29.5} t) = - 4$ by $- 21$ .

$\frac{- 21 \cos (\frac{2 π}{29.5} t)}{- 21} = \frac{- 4}{- 21}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 21$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 21 \cos (\frac{2 π}{29.5} t)}{- 21} = \frac{- 4}{- 21}$

Step 3.2.1.2

Divide $\cos (\frac{2 π}{29.5} t)$ by $1$ .

$\cos (\frac{2 π}{29.5} t) = \frac{- 4}{- 21}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Dividing two negative values results in a positive value.

$\cos (\frac{2 π}{29.5} t) = \frac{4}{21}$

Step 4

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

$\frac{2 π}{29.5} t = \arccos (\frac{4}{21})$

Step 5

Simplify the left side.

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Step 5.1

Simplify $\frac{2 π}{29.5} t$ .

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Step 5.1.1

Replace $π$ with an approximation.

$\frac{2 \cdot 3.14159265}{29.5} t = \arccos (\frac{4}{21})$

Step 5.1.2

Multiply $2$ by $3.14159265$ .

$\frac{6.2831853}{29.5} t = \arccos (\frac{4}{21})$

Step 5.1.3

Divide $6.2831853$ by $29.5$ .

$0.21298933 t = \arccos (\frac{4}{21})$

Step 6

Simplify the right side.

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Step 6.1

Evaluate $\arccos (\frac{4}{21})$ .

$0.21298933 t = 1.37914913$

Step 7

Divide each term in $0.21298933 t = 1.37914913$ by $0.21298933$ and simplify.

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Step 7.1

Divide each term in $0.21298933 t = 1.37914913$ by $0.21298933$ .

$\frac{0.21298933 t}{0.21298933} = \frac{1.37914913}{0.21298933}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $0.21298933$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{0.21298933 t}{0.21298933} = \frac{1.37914913}{0.21298933}$

Step 7.2.1.2

Divide $t$ by $1$ .

$t = \frac{1.37914913}{0.21298933}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $1.37914913$ by $0.21298933$ .

$t = 6.47520284$

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.

$0.21298933 t = 2 (3.14159265) - 1.37914913$

Step 9

Solve for $t$ .

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Step 9.1

Simplify.

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Step 9.1.1

Multiply $2$ by $3.14159265$ .

$0.21298933 t = 6.2831853 - 1.37914913$

Step 9.1.2

Subtract $1.37914913$ from $6.2831853$ .

$0.21298933 t = 4.90403617$

Step 9.2

Divide each term in $0.21298933 t = 4.90403617$ by $0.21298933$ and simplify.

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Step 9.2.1

Divide each term in $0.21298933 t = 4.90403617$ by $0.21298933$ .

$\frac{0.21298933 t}{0.21298933} = \frac{4.90403617}{0.21298933}$

Step 9.2.2

Simplify the left side.

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Step 9.2.2.1

Cancel the common factor of $0.21298933$ .

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Step 9.2.2.1.1

Cancel the common factor.

$\frac{0.21298933 t}{0.21298933} = \frac{4.90403617}{0.21298933}$

Step 9.2.2.1.2

Divide $t$ by $1$ .

$t = \frac{4.90403617}{0.21298933}$

Step 9.2.3

Simplify the right side.

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Step 9.2.3.1

Divide $4.90403617$ by $0.21298933$ .

$t = 23.02479715$

Step 10

Find the period of $\cos (\frac{2 π}{29.5} t)$ .

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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 $0.21298933$ in the formula for period.

$\frac{2 π}{| 0.21298933 |}$

Step 10.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0.21298933$ is $0.21298933$ .

$\frac{2 π}{0.21298933}$

Step 10.4

Replace $π$ with an approximation.

$\frac{2 \cdot 3.14159265}{0.21298933}$

Step 10.5

Multiply $2$ by $3.14159265$ .

$\frac{6.2831853}{0.21298933}$

Step 10.6

Divide $6.2831853$ by $0.21298933$ .

$29.5$

Step 11

The period of the $\cos (\frac{2 π}{29.5} t)$ function is $29.5$ so values will repeat every $29.5$ radians in both directions.

$t = 6.47520284 + 29.5 n, 23.02479715 + 29.5 n$ , for any integer $n$