Trigonometry Examples

$\frac{1}{e^{x} - e^{- x}} = 4$

Step 1

Multiply both sides by $e^{x} - e^{- x}$ .

$\frac{1}{e^{x} - e^{- x}} (e^{x} - e^{- x}) = 4 (e^{x} - e^{- x})$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $e^{x} - e^{- x}$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{1}{e^{x} - e^{- x}} (e^{x} - e^{- x}) = 4 (e^{x} - e^{- x})$

Step 2.1.1.2

Rewrite the expression.

$1 = 4 (e^{x} - e^{- x})$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $4 (e^{x} - e^{- x})$ .

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Step 2.2.1.1

Apply the distributive property.

$1 = 4 e^{x} + 4 (- e^{- x})$

Step 2.2.1.2

Multiply $- 1$ by $4$ .

$1 = 4 e^{x} - 4 e^{- x}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $4 e^{x} - 4 e^{- x} = 1$ .

$4 e^{x} - 4 e^{- x} = 1$

Step 3.2

Rewrite $e^{- x}$ as exponentiation.

$4 e^{x} - 4 {(e^{x})}^{- 1} = 1$

Step 3.3

Substitute $u$ for $e^{x}$ .

$4 u - 4 u^{- 1} = 1$

Step 3.4

Simplify each term.

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Step 3.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 u - 4 \frac{1}{u} = 1$

Step 3.4.2

Combine $- 4$ and $\frac{1}{u}$ .

$4 u + \frac{- 4}{u} = 1$

Step 3.4.3

Move the negative in front of the fraction.

$4 u - \frac{4}{u} = 1$

Step 3.5

Solve for $u$ .

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Step 3.5.1

Find the LCD of the terms in the equation.

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Step 3.5.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, u, 1$

Step 3.5.1.2

The LCM of one and any expression is the expression.

$u$

Step 3.5.2

Multiply each term in $4 u - \frac{4}{u} = 1$ by $u$ to eliminate the fractions.

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Step 3.5.2.1

Multiply each term in $4 u - \frac{4}{u} = 1$ by $u$ .

$4 u \cdot u - \frac{4}{u} u = 1 u$

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Simplify each term.

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Step 3.5.2.2.1.1

Multiply $u$ by $u$ by adding the exponents.

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Step 3.5.2.2.1.1.1

Move $u$ .

$4 (u \cdot u) - \frac{4}{u} u = 1 u$

Step 3.5.2.2.1.1.2

Multiply $u$ by $u$ .

$4 u^{2} - \frac{4}{u} u = 1 u$

Step 3.5.2.2.1.2

Cancel the common factor of $u$ .

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Step 3.5.2.2.1.2.1

Move the leading negative in $- \frac{4}{u}$ into the numerator.

$4 u^{2} + \frac{- 4}{u} u = 1 u$

Step 3.5.2.2.1.2.2

Cancel the common factor.

$4 u^{2} + \frac{- 4}{u} u = 1 u$

Step 3.5.2.2.1.2.3

Rewrite the expression.

$4 u^{2} - 4 = 1 u$

Step 3.5.2.3

Simplify the right side.

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Step 3.5.2.3.1

Multiply $u$ by $1$ .

$4 u^{2} - 4 = u$

Step 3.5.3

Solve the equation.

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Step 3.5.3.1

Subtract $u$ from both sides of the equation.

$4 u^{2} - 4 - u = 0$

Step 3.5.3.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.5.3.3

Substitute the values $a = 4$ , $b = - 1$ , and $c = - 4$ into the quadratic formula and solve for $u$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (4 \cdot - 4)}}{2 \cdot 4}$

Step 3.5.3.4

Simplify.

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Step 3.5.3.4.1

Simplify the numerator.

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Step 3.5.3.4.1.1

Raise $- 1$ to the power of $2$ .

$u = \frac{1 \pm \sqrt{1 - 4 \cdot 4 \cdot - 4}}{2 \cdot 4}$

Step 3.5.3.4.1.2

Multiply $- 4 \cdot 4 \cdot - 4$ .

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Step 3.5.3.4.1.2.1

Multiply $- 4$ by $4$ .

$u = \frac{1 \pm \sqrt{1 - 16 \cdot - 4}}{2 \cdot 4}$

Step 3.5.3.4.1.2.2

Multiply $- 16$ by $- 4$ .

$u = \frac{1 \pm \sqrt{1 + 64}}{2 \cdot 4}$

Step 3.5.3.4.1.3

Add $1$ and $64$ .

$u = \frac{1 \pm \sqrt{65}}{2 \cdot 4}$

Step 3.5.3.4.2

Multiply $2$ by $4$ .

$u = \frac{1 \pm \sqrt{65}}{8}$

Step 3.5.3.5

The final answer is the combination of both solutions.

$u = \frac{1 + \sqrt{65}}{8}, \frac{1 - \sqrt{65}}{8}$

Step 3.6

Substitute $\frac{1 + \sqrt{65}}{8}$ for $u$ in $u = e^{x}$ .

$\frac{1 + \sqrt{65}}{8} = e^{x}$

Step 3.7

Solve $\frac{1 + \sqrt{65}}{8} = e^{x}$ .

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Step 3.7.1

Rewrite the equation as $e^{x} = \frac{1 + \sqrt{65}}{8}$ .

$e^{x} = \frac{1 + \sqrt{65}}{8}$

Step 3.7.2

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

$\ln (e^{x}) = \ln (\frac{1 + \sqrt{65}}{8})$

Step 3.7.3

Expand the left side.

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Step 3.7.3.1

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

$x \ln (e) = \ln (\frac{1 + \sqrt{65}}{8})$

Step 3.7.3.2

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

$x \cdot 1 = \ln (\frac{1 + \sqrt{65}}{8})$

Step 3.7.3.3

Multiply $x$ by $1$ .

$x = \ln (\frac{1 + \sqrt{65}}{8})$

Step 3.8

Substitute $\frac{1 - \sqrt{65}}{8}$ for $u$ in $u = e^{x}$ .

$\frac{1 - \sqrt{65}}{8} = e^{x}$

Step 3.9

Solve $\frac{1 - \sqrt{65}}{8} = e^{x}$ .

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Step 3.9.1

Rewrite the equation as $e^{x} = \frac{1 - \sqrt{65}}{8}$ .

$e^{x} = \frac{1 - \sqrt{65}}{8}$

Step 3.9.2

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

$\ln (e^{x}) = \ln (\frac{1 - \sqrt{65}}{8})$

Step 3.9.3

The equation cannot be solved because $\ln (\frac{1 - \sqrt{65}}{8})$ is undefined.

Undefined

Step 3.9.4

There is no solution for $e^{x} = \frac{1 - \sqrt{65}}{8}$

No solution

Step 3.10

List the solutions that makes the equation true.

$x = \ln (\frac{1 + \sqrt{65}}{8})$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \ln (\frac{1 + \sqrt{65}}{8})$

Decimal Form:

$x = 0.12467674 \dots$