Trigonometry Examples

$d = 3 \cos (π t - \frac{π}{2})$

Step 1

Rewrite the equation as $3 \cos (π t - \frac{π}{2}) = d$ .

$3 \cos (π t - \frac{π}{2}) = d$

Step 2

Divide each term in $3 \cos (π t - \frac{π}{2}) = d$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 \cos (π t - \frac{π}{2}) = d$ by $3$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

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

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

Step 3

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

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

Step 4

Add $\frac{π}{2}$ to both sides of the equation.

$π t = \arccos (\frac{d}{3}) + \frac{π}{2}$

Step 5

Divide each term in $π t = \arccos (\frac{d}{3}) + \frac{π}{2}$ by $π$ and simplify.

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Step 5.1

Divide each term in $π t = \arccos (\frac{d}{3}) + \frac{π}{2}$ by $π$ .

$\frac{π t}{π} = \frac{\arccos (\frac{d}{3})}{π} + \frac{\frac{π}{2}}{π}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $π$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{π t}{π} = \frac{\arccos (\frac{d}{3})}{π} + \frac{\frac{π}{2}}{π}$

Step 5.2.1.2

Divide $t$ by $1$ .

$t = \frac{\arccos (\frac{d}{3})}{π} + \frac{\frac{π}{2}}{π}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$t = \frac{\arccos (\frac{d}{3})}{π} + \frac{π}{2} \cdot \frac{1}{π}$

Step 5.3.1.2

Cancel the common factor of $π$ .

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Step 5.3.1.2.1

Cancel the common factor.

$t = \frac{\arccos (\frac{d}{3})}{π} + \frac{π}{2} \cdot \frac{1}{π}$

Step 5.3.1.2.2

Rewrite the expression.

$t = \frac{\arccos (\frac{d}{3})}{π} + \frac{1}{2}$