Trigonometry Examples

$i = 5.5 (1 - e^{- 0.23 t \cdot 0.006})$

Step 1

Since $t$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$5.5 (1 - e^{- 0.23 t \cdot 0.006}) = i$

Step 2

Multiply $0.006$ by $- 0.23$ .

$5.5 (1 - e^{- 0.00138 t}) = i$

Step 3

Divide each term in $5.5 (1 - e^{- 0.00138 t}) = i$ by $5.5$ and simplify.

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Step 3.1

Divide each term in $5.5 (1 - e^{- 0.00138 t}) = i$ by $5.5$ .

$\frac{5.5 (1 - e^{- 0.00138 t})}{5.5} = \frac{i}{5.5}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $5.5$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{5.5 (1 - e^{- 0.00138 t})}{5.5} = \frac{i}{5.5}$

Step 3.2.1.2

Divide $1 - e^{- 0.00138 t}$ by $1$ .

$1 - e^{- 0.00138 t} = \frac{i}{5.5}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply by $1$ .

$1 - e^{- 0.00138 t} = \frac{1 i}{5.5}$

Step 3.3.2

Factor $5.5$ out of $5.5$ .

$1 - e^{- 0.00138 t} = \frac{1 i}{5.5 (1)}$

Step 3.3.3

Separate fractions.

$1 - e^{- 0.00138 t} = \frac{1}{5.5} \cdot \frac{i}{1}$

Step 3.3.4

Divide $1$ by $5.5$ .

$1 - e^{- 0.00138 t} = 0.\overline{18} \frac{i}{1}$

Step 3.3.5

Divide $i$ by $1$ .

$1 - e^{- 0.00138 t} = 0.\overline{18} i$

Step 4

Move all terms not containing $t$ to the right side of the equation.

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Step 4.1

Subtract $1$ from both sides of the equation.

$- e^{- 0.00138 t} = 0.\overline{18} i - 1$

Step 4.2

Reorder $0.\overline{18} i$ and $- 1$ .

$- e^{- 0.00138 t} = - 1 + 0.\overline{18} i$

Step 5

Divide each term in $- e^{- 0.00138 t} = - 1 + 0.\overline{18} i$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- e^{- 0.00138 t} = - 1 + 0.\overline{18} i$ by $- 1$ .

$\frac{- e^{- 0.00138 t}}{- 1} = \frac{- 1}{- 1} + \frac{0.\overline{18} i}{- 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{e^{- 0.00138 t}}{1} = \frac{- 1}{- 1} + \frac{0.\overline{18} i}{- 1}$

Step 5.2.2

Divide $e^{- 0.00138 t}$ by $1$ .

$e^{- 0.00138 t} = \frac{- 1}{- 1} + \frac{0.\overline{18} i}{- 1}$

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

Divide $- 1$ by $- 1$ .

$e^{- 0.00138 t} = 1 + \frac{0.\overline{18} i}{- 1}$

Step 5.3.1.2

Move the negative one from the denominator of $\frac{0.\overline{18} i}{- 1}$ .

$e^{- 0.00138 t} = 1 - 1 \cdot (0.\overline{18} i)$

Step 5.3.1.3

Rewrite $- 1 \cdot (0.\overline{18} i)$ as $- (0.\overline{18} i)$ .

$e^{- 0.00138 t} = 1 - (0.\overline{18} i)$

Step 5.3.1.4

Multiply $0.\overline{18}$ by $- 1$ .

$e^{- 0.00138 t} = 1 - 0.\overline{18} i$

Step 6

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{- 0.00138 t}) = \ln (1 - 0.\overline{18} i)$

Step 7

Expand the left side.

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Step 7.1

Expand $\ln (e^{- 0.00138 t})$ by moving $- 0.00138 t$ outside the logarithm.

$- 0.00138 t \ln (e) = \ln (1 - 0.\overline{18} i)$

Step 7.2

The natural logarithm of $e$ is $1$ .

$- 0.00138 t \cdot 1 = \ln (1 - 0.\overline{18} i)$

Step 7.3

Multiply $- 0.00138$ by $1$ .

$- 0.00138 t = \ln (1 - 0.\overline{18} i)$

Step 8

Divide each term in $- 0.00138 t = \ln (1 - 0.\overline{18} i)$ by $- 0.00138$ and simplify.

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Step 8.1

Divide each term in $- 0.00138 t = \ln (1 - 0.\overline{18} i)$ by $- 0.00138$ .

$\frac{- 0.00138 t}{- 0.00138} = \frac{\ln (1 - 0.\overline{18} i)}{- 0.00138}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $- 0.00138$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{- 0.00138 t}{- 0.00138} = \frac{\ln (1 - 0.\overline{18} i)}{- 0.00138}$

Step 8.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (1 - 0.\overline{18} i)}{- 0.00138}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (1 - 0.\overline{18} i)}{0.00138}$

Step 8.3.2

Multiply by $1$ .

$t = - \frac{1 \ln (1 - 0.\overline{18} i)}{0.00138}$

Step 8.3.3

Factor $0.00138$ out of $0.00138$ .

$t = - \frac{1 \ln (1 - 0.\overline{18} i)}{0.00138 (1)}$

Step 8.3.4

Separate fractions.

$t = - (\frac{1}{0.00138} \cdot \frac{\ln (1 - 0.\overline{18} i)}{1})$

Step 8.3.5

Divide $1$ by $0.00138$ .

$t = - (724.63768115 \frac{\ln (1 - 0.\overline{18} i)}{1})$

Step 8.3.6

Divide $\ln (1 - 0.\overline{18} i)$ by $1$ .

$t = - (724.63768115 \ln (1 - 0.\overline{18} i))$

Step 8.3.7

Multiply $724.63768115$ by $- 1$ .

$t = - 724.63768115 \ln (1 - 0.\overline{18} i)$