Trigonometry Examples

$\frac{e^{x} + e^{- x}}{3} = 10$

Step 1

Multiply both sides by $3$ .

$\frac{e^{x} + e^{- x}}{3} \cdot 3 = 10 \cdot 3$

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 $3$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{e^{x} + e^{- x}}{3} \cdot 3 = 10 \cdot 3$

Step 2.1.1.2

Rewrite the expression.

$e^{x} + e^{- x} = 10 \cdot 3$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $10$ by $3$ .

$e^{x} + e^{- x} = 30$

Step 3

Solve for $x$ .

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Step 3.1

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

$e^{x} + {(e^{x})}^{- 1} = 30$

Step 3.2

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

$u + u^{- 1} = 30$

Step 3.3

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

$u + \frac{1}{u} = 30$

Step 3.4

Solve for $u$ .

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Step 3.4.1

Find the LCD of the terms in the equation.

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Step 3.4.1.1

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 3.4.1.2

The LCM of one and any expression is the expression.

$u$

Step 3.4.2

Multiply each term in $u + \frac{1}{u} = 30$ by $u$ to eliminate the fractions.

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Step 3.4.2.1

Multiply each term in $u + \frac{1}{u} = 30$ by $u$ .

$u \cdot u + \frac{1}{u} u = 30 u$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Simplify each term.

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Step 3.4.2.2.1.1

Multiply $u$ by $u$ .

$u^{2} + \frac{1}{u} u = 30 u$

Step 3.4.2.2.1.2

Cancel the common factor of $u$ .

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Step 3.4.2.2.1.2.1

Cancel the common factor.

$u^{2} + \frac{1}{u} u = 30 u$

Step 3.4.2.2.1.2.2

Rewrite the expression.

$u^{2} + 1 = 30 u$

Step 3.4.3

Solve the equation.

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Step 3.4.3.1

Subtract $30 u$ from both sides of the equation.

$u^{2} + 1 - 30 u = 0$

Step 3.4.3.2

Use the quadratic formula to find the solutions.

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

Step 3.4.3.3

Substitute the values $a = 1$ , $b = - 30$ , and $c = 1$ into the quadratic formula and solve for $u$ .

$\frac{30 \pm \sqrt{{(- 30)}^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 3.4.3.4

Simplify.

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Step 3.4.3.4.1

Simplify the numerator.

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Step 3.4.3.4.1.1

Raise $- 30$ to the power of $2$ .

$u = \frac{30 \pm \sqrt{900 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 3.4.3.4.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 3.4.3.4.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{30 \pm \sqrt{900 - 4 \cdot 1}}{2 \cdot 1}$

Step 3.4.3.4.1.2.2

Multiply $- 4$ by $1$ .

$u = \frac{30 \pm \sqrt{900 - 4}}{2 \cdot 1}$

Step 3.4.3.4.1.3

Subtract $4$ from $900$ .

$u = \frac{30 \pm \sqrt{896}}{2 \cdot 1}$

Step 3.4.3.4.1.4

Rewrite $896$ as $8^{2} \cdot 14$ .

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Step 3.4.3.4.1.4.1

Factor $64$ out of $896$ .

$u = \frac{30 \pm \sqrt{64 (14)}}{2 \cdot 1}$

Step 3.4.3.4.1.4.2

Rewrite $64$ as $8^{2}$ .

$u = \frac{30 \pm \sqrt{8^{2} \cdot 14}}{2 \cdot 1}$

Step 3.4.3.4.1.5

Pull terms out from under the radical.

$u = \frac{30 \pm 8 \sqrt{14}}{2 \cdot 1}$

Step 3.4.3.4.2

Multiply $2$ by $1$ .

$u = \frac{30 \pm 8 \sqrt{14}}{2}$

Step 3.4.3.4.3

Simplify $\frac{30 \pm 8 \sqrt{14}}{2}$ .

$u = 15 \pm 4 \sqrt{14}$

Step 3.4.3.5

The final answer is the combination of both solutions.

$u = 15 + 4 \sqrt{14}, 15 - 4 \sqrt{14}$

Step 3.5

Substitute $15 + 4 \sqrt{14}$ for $u$ in $u = e^{x}$ .

$15 + 4 \sqrt{14} = e^{x}$

Step 3.6

Solve $15 + 4 \sqrt{14} = e^{x}$ .

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Step 3.6.1

Rewrite the equation as $e^{x} = 15 + 4 \sqrt{14}$ .

$e^{x} = 15 + 4 \sqrt{14}$

Step 3.6.2

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

$\ln (e^{x}) = \ln (15 + 4 \sqrt{14})$

Step 3.6.3

Expand the left side.

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Step 3.6.3.1

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

$x \ln (e) = \ln (15 + 4 \sqrt{14})$

Step 3.6.3.2

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

$x \cdot 1 = \ln (15 + 4 \sqrt{14})$

Step 3.6.3.3

Multiply $x$ by $1$ .

$x = \ln (15 + 4 \sqrt{14})$

Step 3.7

Substitute $15 - 4 \sqrt{14}$ for $u$ in $u = e^{x}$ .

$15 - 4 \sqrt{14} = e^{x}$

Step 3.8

Solve $15 - 4 \sqrt{14} = e^{x}$ .

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Step 3.8.1

Rewrite the equation as $e^{x} = 15 - 4 \sqrt{14}$ .

$e^{x} = 15 - 4 \sqrt{14}$

Step 3.8.2

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

$\ln (e^{x}) = \ln (15 - 4 \sqrt{14})$

Step 3.8.3

Expand the left side.

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Step 3.8.3.1

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

$x \ln (e) = \ln (15 - 4 \sqrt{14})$

Step 3.8.3.2

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

$x \cdot 1 = \ln (15 - 4 \sqrt{14})$

Step 3.8.3.3

Multiply $x$ by $1$ .

$x = \ln (15 - 4 \sqrt{14})$

Step 3.9

List the solutions that makes the equation true.

$x = \ln (15 + 4 \sqrt{14}), \ln (15 - 4 \sqrt{14})$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \ln (15 + 4 \sqrt{14}), \ln (15 - 4 \sqrt{14})$

Decimal Form:

$x = 3.40008441 \dots, - 3.40008441 \dots$