Precalculus Examples

$d = 8 \cos (\frac{3 π}{4} t)$

Step 1

Rewrite the equation as $8 \cos (\frac{3 π}{4} t) = d$ .

$8 \cos (\frac{3 π}{4} t) = d$

Step 2

Divide each term in $8 \cos (\frac{3 π}{4} t) = d$ by $8$ and simplify.

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Step 2.1

Divide each term in $8 \cos (\frac{3 π}{4} t) = d$ by $8$ .

$\frac{8 \cos (\frac{3 π}{4} t)}{8} = \frac{d}{8}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $8$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{8 \cos (\frac{3 π}{4} t)}{8} = \frac{d}{8}$

Step 2.2.1.2

Divide $\cos (\frac{3 π}{4} t)$ by $1$ .

$\cos (\frac{3 π}{4} t) = \frac{d}{8}$

Step 3

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

$\frac{3 π}{4} t = \arccos (\frac{d}{8})$

Step 4

Simplify the left side.

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Step 4.1

Combine $\frac{3 π}{4}$ and $t$ .

$\frac{3 π t}{4} = \arccos (\frac{d}{8})$

Step 5

Multiply both sides of the equation by $\frac{4}{3 π}$ .

$\frac{4}{3 π} \cdot \frac{3 π t}{4} = \frac{4}{3 π} \arccos (\frac{d}{8})$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $\frac{4}{3 π} \cdot \frac{3 π t}{4}$ .

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Step 6.1.1.1

Cancel the common factor of $4$ .

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Step 6.1.1.1.1

Cancel the common factor.

$\frac{4}{3 π} \cdot \frac{3 π t}{4} = \frac{4}{3 π} \arccos (\frac{d}{8})$

Step 6.1.1.1.2

Rewrite the expression.

$\frac{1}{3 π} (3 π t) = \frac{4}{3 π} \arccos (\frac{d}{8})$

Step 6.1.1.2

Cancel the common factor of $3 π$ .

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Step 6.1.1.2.1

Factor $3 π$ out of $3 π t$ .

$\frac{1}{3 π} (3 π (t)) = \frac{4}{3 π} \arccos (\frac{d}{8})$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{1}{3 π} (3 π t) = \frac{4}{3 π} \arccos (\frac{d}{8})$

Step 6.1.1.2.3

Rewrite the expression.

$t = \frac{4}{3 π} \arccos (\frac{d}{8})$

Step 6.2

Simplify the right side.

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Step 6.2.1

Combine $\frac{4}{3 π}$ and $\arccos (\frac{d}{8})$ .

$t = \frac{4 \arccos (\frac{d}{8})}{3 π}$