Trigonometry Examples

$\frac{10^{x} + 10^{- x}}{10^{x} - 10^{- x}} = 6$

Step 1

Multiply both sides by $10^{x} - 10^{- x}$ .

$\frac{10^{x} + 10^{- x}}{10^{x} - 10^{- x}} (10^{x} - 10^{- x}) = 6 (10^{x} - 10^{- x})$

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 $10^{x} - 10^{- x}$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{10^{x} + 10^{- x}}{10^{x} - 10^{- x}} (10^{x} - 10^{- x}) = 6 (10^{x} - 10^{- x})$

Step 2.1.1.2

Rewrite the expression.

$10^{x} + 10^{- x} = 6 (10^{x} - 10^{- x})$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $6 (10^{x} - 10^{- x})$ .

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Step 2.2.1.1

Apply the distributive property.

$10^{x} + 10^{- x} = 6 \cdot 10^{x} + 6 (- 10^{- x})$

Step 2.2.1.2

Multiply $- 1$ by $6$ .

$10^{x} + 10^{- x} = 6 \cdot 10^{x} - 6 \cdot 10^{- x}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite $10^{- x}$ as exponentiation.

$10^{x} + {(10^{x})}^{- 1} = 6 \cdot 10^{x} - 6 \cdot 10^{- x}$

Step 3.2

Substitute $u$ for $10^{x}$ .

$u + u^{- 1} = 6 \cdot 10^{x} - 6 \cdot 10^{- x}$

Step 3.3

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

$u + \frac{1}{u} = 6 \cdot 10^{x} - 6 \cdot 10^{- x}$

Step 3.4

Rewrite $10^{- x}$ as exponentiation.

$u + \frac{1}{u} = 6 \cdot 10^{x} - 6 \cdot {(10^{x})}^{- 1} = 6 \cdot 10^{x} - 6 \cdot 10^{- x}$

Step 3.5

Substitute $u$ for $10^{x}$ .

$u + \frac{1}{u} = 6 \cdot u - 6 \cdot u^{- 1} = 6 \cdot 10^{x} - 6 \cdot 10^{- x}$

Step 3.6

Simplify each term.

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Step 3.6.1

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

$u + \frac{1}{u} = 6 u - 6 \cdot \frac{1}{u} = 6 \cdot 10^{x} - 6 \cdot 10^{- x}$

Step 3.6.2

Combine $- 6$ and $\frac{1}{u}$ .

$u + \frac{1}{u} = 6 u + \frac{- 6}{u} = 6 \cdot 10^{x} - 6 \cdot 10^{- x}$

Step 3.6.3

Move the negative in front of the fraction.

$u + \frac{1}{u} = 6 u - \frac{6}{u} = 6 \cdot 10^{x} - 6 \cdot 10^{- x}$

Step 3.7

Move all terms containing $x$ to the left side of the equation.

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Step 3.7.1

Subtract $6 \cdot 10^{x}$ from both sides of the equation.

$10^{x} + 10^{- x} - 6 \cdot 10^{x} = - 6 \cdot 10^{- x}$

Step 3.7.2

Add $6 \cdot 10^{- x}$ to both sides of the equation.

$10^{x} + 10^{- x} - 6 \cdot 10^{x} + 6 \cdot 10^{- x} = 0$

Step 3.7.3

Subtract $6 \cdot 10^{x}$ from $10^{x}$ .

$10^{- x} - 5 \cdot 10^{x} + 6 \cdot 10^{- x} = 0$

Step 3.7.4

Add $10^{- x}$ and $6 \cdot 10^{- x}$ .

$7 \cdot 10^{- x} - 5 \cdot 10^{x} = 0$

Step 3.8

Rewrite $10^{- x}$ as exponentiation.

$7 \cdot {(10^{x})}^{- 1} - 5 \cdot 10^{x} = 0$

Step 3.9

Substitute $u$ for $10^{x}$ .

$7 \cdot u^{- 1} - 5 \cdot u = 0$

Step 3.10

Simplify each term.

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Step 3.10.1

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

$7 \cdot \frac{1}{u} - 5 \cdot u = 0$

Step 3.10.2

Combine $7$ and $\frac{1}{u}$ .

$\frac{7}{u} - 5 u = 0$

Step 3.11

Reorder $\frac{7}{u}$ and $- 5 u$ .

$- 5 u + \frac{7}{u} = 0$

Step 3.12

Solve for $u$ .

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Step 3.12.1

Find the LCD of the terms in the equation.

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Step 3.12.1.1

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 3.12.1.2

The LCM of one and any expression is the expression.

$u$

Step 3.12.2

Multiply each term in $- 5 u + \frac{7}{u} = 0$ by $u$ to eliminate the fractions.

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Step 3.12.2.1

Multiply each term in $- 5 u + \frac{7}{u} = 0$ by $u$ .

$- 5 u \cdot u + \frac{7}{u} u = 0 u$

Step 3.12.2.2

Simplify the left side.

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Step 3.12.2.2.1

Simplify each term.

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Step 3.12.2.2.1.1

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

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Step 3.12.2.2.1.1.1

Move $u$ .

$- 5 (u \cdot u) + \frac{7}{u} u = 0 u$

Step 3.12.2.2.1.1.2

Multiply $u$ by $u$ .

$- 5 u^{2} + \frac{7}{u} u = 0 u$

Step 3.12.2.2.1.2

Cancel the common factor of $u$ .

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Step 3.12.2.2.1.2.1

Cancel the common factor.

$- 5 u^{2} + \frac{7}{u} u = 0 u$

Step 3.12.2.2.1.2.2

Rewrite the expression.

$- 5 u^{2} + 7 = 0 u$

Step 3.12.2.3

Simplify the right side.

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Step 3.12.2.3.1

Multiply $0$ by $u$ .

$- 5 u^{2} + 7 = 0$

Step 3.12.3

Solve the equation.

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Step 3.12.3.1

Subtract $7$ from both sides of the equation.

$- 5 u^{2} = - 7$

Step 3.12.3.2

Divide each term in $- 5 u^{2} = - 7$ by $- 5$ and simplify.

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Step 3.12.3.2.1

Divide each term in $- 5 u^{2} = - 7$ by $- 5$ .

$\frac{- 5 u^{2}}{- 5} = \frac{- 7}{- 5}$

Step 3.12.3.2.2

Simplify the left side.

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Step 3.12.3.2.2.1

Cancel the common factor of $- 5$ .

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Step 3.12.3.2.2.1.1

Cancel the common factor.

$\frac{- 5 u^{2}}{- 5} = \frac{- 7}{- 5}$

Step 3.12.3.2.2.1.2

Divide $u^{2}$ by $1$ .

$u^{2} = \frac{- 7}{- 5}$

Step 3.12.3.2.3

Simplify the right side.

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Step 3.12.3.2.3.1

Dividing two negative values results in a positive value.

$u^{2} = \frac{7}{5}$

Step 3.12.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$u = \pm \sqrt{\frac{7}{5}}$

Step 3.12.3.4

Simplify $\pm \sqrt{\frac{7}{5}}$ .

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Step 3.12.3.4.1

Rewrite $\sqrt{\frac{7}{5}}$ as $\frac{\sqrt{7}}{\sqrt{5}}$ .

$u = \pm \frac{\sqrt{7}}{\sqrt{5}}$

Step 3.12.3.4.2

Multiply $\frac{\sqrt{7}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$u = \pm \frac{\sqrt{7}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 3.12.3.4.3

Combine and simplify the denominator.

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Step 3.12.3.4.3.1

Multiply $\frac{\sqrt{7}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$u = \pm \frac{\sqrt{7} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 3.12.3.4.3.2

Raise $\sqrt{5}$ to the power of $1$ .

$u = \pm \frac{\sqrt{7} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 3.12.3.4.3.3

Raise $\sqrt{5}$ to the power of $1$ .

$u = \pm \frac{\sqrt{7} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 3.12.3.4.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$u = \pm \frac{\sqrt{7} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 3.12.3.4.3.5

Add $1$ and $1$ .

$u = \pm \frac{\sqrt{7} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 3.12.3.4.3.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.12.3.4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$u = \pm \frac{\sqrt{7} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 3.12.3.4.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$u = \pm \frac{\sqrt{7} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 3.12.3.4.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$u = \pm \frac{\sqrt{7} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 3.12.3.4.3.6.4

Cancel the common factor of $2$ .

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Step 3.12.3.4.3.6.4.1

Cancel the common factor.

$u = \pm \frac{\sqrt{7} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 3.12.3.4.3.6.4.2

Rewrite the expression.

$u = \pm \frac{\sqrt{7} \sqrt{5}}{5^{1}}$

Step 3.12.3.4.3.6.5

Evaluate the exponent.

$u = \pm \frac{\sqrt{7} \sqrt{5}}{5}$

Step 3.12.3.4.4

Simplify the numerator.

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Step 3.12.3.4.4.1

Combine using the product rule for radicals.

$u = \pm \frac{\sqrt{7 \cdot 5}}{5}$

Step 3.12.3.4.4.2

Multiply $7$ by $5$ .

$u = \pm \frac{\sqrt{35}}{5}$

Step 3.12.3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.12.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$u = \frac{\sqrt{35}}{5}$

Step 3.12.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$u = - \frac{\sqrt{35}}{5}$

Step 3.12.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$u = \frac{\sqrt{35}}{5}, - \frac{\sqrt{35}}{5}$

Step 3.13

Substitute $\frac{\sqrt{35}}{5}$ for $u$ in $u = 10^{x}$ .

$\frac{\sqrt{35}}{5} = 10^{x}$

Step 3.14

Solve $\frac{\sqrt{35}}{5} = 10^{x}$ .

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Step 3.14.1

Rewrite the equation as $10^{x} = \frac{\sqrt{35}}{5}$ .

$10^{x} = \frac{\sqrt{35}}{5}$

Step 3.14.2

Take the base $10$ logarithm of both sides of the equation to remove the variable from the exponent.

$\log (10^{x}) = \log (\frac{\sqrt{35}}{5})$

Step 3.14.3

Expand the left side.

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Step 3.14.3.1

Expand $\log (10^{x})$ by moving $x$ outside the logarithm.

$x \log (10) = \log (\frac{\sqrt{35}}{5})$

Step 3.14.3.2

Logarithm base $10$ of $10$ is $1$ .

$x \cdot 1 = \log (\frac{\sqrt{35}}{5})$

Step 3.14.3.3

Multiply $x$ by $1$ .

$x = \log (\frac{\sqrt{35}}{5})$

Step 3.15

Substitute $- \frac{\sqrt{35}}{5}$ for $u$ in $u = 10^{x}$ .

$- \frac{\sqrt{35}}{5} = 10^{x}$

Step 3.16

Solve $- \frac{\sqrt{35}}{5} = 10^{x}$ .

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Step 3.16.1

Rewrite the equation as $10^{x} = - \frac{\sqrt{35}}{5}$ .

$10^{x} = - \frac{\sqrt{35}}{5}$

Step 3.16.2

Take the base $10$ logarithm of both sides of the equation to remove the variable from the exponent.

$\log (10^{x}) = \log (- \frac{\sqrt{35}}{5})$

Step 3.16.3

The equation cannot be solved because $\log (- \frac{\sqrt{35}}{5})$ is undefined.

Undefined

Step 3.16.4

There is no solution for $10^{x} = - \frac{\sqrt{35}}{5}$

No solution

Step 3.17

List the solutions that makes the equation true.

$x = \log (\frac{\sqrt{35}}{5})$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \log (\frac{\sqrt{35}}{5})$

Decimal Form:

$x = 0.07306401 \dots$