Trigonometry Examples

$h = - 2.5 \cos (\frac{4 π}{3} t) + 2.5$

Step 1

Rewrite the equation as $- 2.5 \cos (\frac{4 π}{3} t) + 2.5 = h$ .

$- 2.5 \cos (\frac{4 π}{3} t) + 2.5 = h$

Step 2

Subtract $2.5$ from both sides of the equation.

$- 2.5 \cos (\frac{4 π}{3} t) = h - 2.5$

Step 3

Divide each term in $- 2.5 \cos (\frac{4 π}{3} t) = h - 2.5$ by $- 2.5$ and simplify.

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Step 3.1

Divide each term in $- 2.5 \cos (\frac{4 π}{3} t) = h - 2.5$ by $- 2.5$ .

$\frac{- 2.5 \cos (\frac{4 π}{3} t)}{- 2.5} = \frac{h}{- 2.5} + \frac{- 2.5}{- 2.5}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 2.5$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 2.5 \cos (\frac{4 π}{3} t)}{- 2.5} = \frac{h}{- 2.5} + \frac{- 2.5}{- 2.5}$

Step 3.2.1.2

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

$\cos (\frac{4 π}{3} t) = \frac{h}{- 2.5} + \frac{- 2.5}{- 2.5}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Move the negative in front of the fraction.

$\cos (\frac{4 π}{3} t) = - \frac{h}{2.5} + \frac{- 2.5}{- 2.5}$

Step 3.3.1.2

Multiply by $1$ .

$\cos (\frac{4 π}{3} t) = - \frac{1 h}{2.5} + \frac{- 2.5}{- 2.5}$

Step 3.3.1.3

Factor $2.5$ out of $2.5$ .

$\cos (\frac{4 π}{3} t) = - \frac{1 h}{2.5 (1)} + \frac{- 2.5}{- 2.5}$

Step 3.3.1.4

Separate fractions.

$\cos (\frac{4 π}{3} t) = - (\frac{1}{2.5} \cdot \frac{h}{1}) + \frac{- 2.5}{- 2.5}$

Step 3.3.1.5

Divide $1$ by $2.5$ .

$\cos (\frac{4 π}{3} t) = - (0.4 \frac{h}{1}) + \frac{- 2.5}{- 2.5}$

Step 3.3.1.6

Divide $h$ by $1$ .

$\cos (\frac{4 π}{3} t) = - (0.4 h) + \frac{- 2.5}{- 2.5}$

Step 3.3.1.7

Multiply $0.4$ by $- 1$ .

$\cos (\frac{4 π}{3} t) = - 0.4 h + \frac{- 2.5}{- 2.5}$

Step 3.3.1.8

Divide $- 2.5$ by $- 2.5$ .

$\cos (\frac{4 π}{3} t) = - 0.4 h + 1$

Step 4

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

$\frac{4 π}{3} t = \arccos (- 0.4 h + 1)$

Step 5

Simplify the left side.

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Step 5.1

Combine $\frac{4 π}{3}$ and $t$ .

$\frac{4 π t}{3} = \arccos (- 0.4 h + 1)$

Step 6

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

$\frac{3}{4 π} \cdot \frac{4 π t}{3} = \frac{3}{4 π} \arccos (- 0.4 h + 1)$

Step 7

Simplify both sides of the equation.

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Step 7.1

Simplify the left side.

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Step 7.1.1

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

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Step 7.1.1.1

Cancel the common factor of $3$ .

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Step 7.1.1.1.1

Cancel the common factor.

$\frac{3}{4 π} \cdot \frac{4 π t}{3} = \frac{3}{4 π} \arccos (- 0.4 h + 1)$

Step 7.1.1.1.2

Rewrite the expression.

$\frac{1}{4 π} (4 π t) = \frac{3}{4 π} \arccos (- 0.4 h + 1)$

Step 7.1.1.2

Cancel the common factor of $4 π$ .

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Step 7.1.1.2.1

Factor $4 π$ out of $4 π t$ .

$\frac{1}{4 π} (4 π (t)) = \frac{3}{4 π} \arccos (- 0.4 h + 1)$

Step 7.1.1.2.2

Cancel the common factor.

$\frac{1}{4 π} (4 π t) = \frac{3}{4 π} \arccos (- 0.4 h + 1)$

Step 7.1.1.2.3

Rewrite the expression.

$t = \frac{3}{4 π} \arccos (- 0.4 h + 1)$

Step 7.2

Simplify the right side.

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Step 7.2.1

Combine $\frac{3}{4 π}$ and $\arccos (- 0.4 h + 1)$ .

$t = \frac{3 \arccos (- 0.4 h + 1)}{4 π}$