Trigonometry Examples

$| \frac{10 x}{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})} | = 1$

Step 1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$\frac{10 x}{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})} = \pm 1$

Step 2

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

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Step 2.1

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

$\frac{10 x}{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})} = 1$

Step 2.2

Multiply both sides of the equation by $\frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10}$ .

$\frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot \frac{10 x}{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})} = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot 1$

Step 2.3

Simplify both sides of the equation.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Simplify $\frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot \frac{10 x}{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}$ .

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Step 2.3.1.1.1

Combine.

$\frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46}) (10 x)}{10 ((2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46}))} = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot 1$

Step 2.3.1.1.2

Cancel the common factor of $2 + i^{3.46}$ .

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Step 2.3.1.1.2.1

Cancel the common factor.

$\frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46}) (10 x)}{10 ((2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46}))} = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot 1$

Step 2.3.1.1.2.2

Rewrite the expression.

$\frac{((12 + i^{3.46}) (66.6 + i^{3.46})) (10 x)}{10 ((12 + i^{3.46}) (66.6 + i^{3.46}))} = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot 1$

Step 2.3.1.1.3

Cancel the common factor of $12 + i^{3.46}$ .

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Step 2.3.1.1.3.1

Cancel the common factor.

$\frac{(12 + i^{3.46}) (66.6 + i^{3.46}) (10 x)}{10 ((12 + i^{3.46}) (66.6 + i^{3.46}))} = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot 1$

Step 2.3.1.1.3.2

Rewrite the expression.

$\frac{(66.6 + i^{3.46}) (10 x)}{10 (66.6 + i^{3.46})} = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot 1$

Step 2.3.1.1.4

Cancel the common factor of $66.6 + i^{3.46}$ .

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Step 2.3.1.1.4.1

Cancel the common factor.

$\frac{(66.6 + i^{3.46}) (10 x)}{10 (66.6 + i^{3.46})} = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot 1$

Step 2.3.1.1.4.2

Rewrite the expression.

$\frac{10 x}{10} = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot 1$

Step 2.3.1.1.5

Cancel the common factor of $10$ .

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Step 2.3.1.1.5.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot 1$

Step 2.3.1.1.5.2

Divide $x$ by $1$ .

$x = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot 1$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $\frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10}$ by $1$ .

$x = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10}$

Step 2.4

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

$\frac{10 x}{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})} = - 1$

Step 2.5

Multiply both sides of the equation by $\frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10}$ .

$\frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot \frac{10 x}{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})} = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot - 1$

Step 2.6

Simplify both sides of the equation.

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Step 2.6.1

Simplify the left side.

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Step 2.6.1.1

Simplify $\frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot \frac{10 x}{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}$ .

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Step 2.6.1.1.1

Combine.

$\frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46}) (10 x)}{10 ((2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46}))} = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot - 1$

Step 2.6.1.1.2

Cancel the common factor of $2 + i^{3.46}$ .

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Step 2.6.1.1.2.1

Cancel the common factor.

$\frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46}) (10 x)}{10 ((2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46}))} = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot - 1$

Step 2.6.1.1.2.2

Rewrite the expression.

$\frac{((12 + i^{3.46}) (66.6 + i^{3.46})) (10 x)}{10 ((12 + i^{3.46}) (66.6 + i^{3.46}))} = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot - 1$

Step 2.6.1.1.3

Cancel the common factor of $12 + i^{3.46}$ .

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Step 2.6.1.1.3.1

Cancel the common factor.

$\frac{(12 + i^{3.46}) (66.6 + i^{3.46}) (10 x)}{10 ((12 + i^{3.46}) (66.6 + i^{3.46}))} = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot - 1$

Step 2.6.1.1.3.2

Rewrite the expression.

$\frac{(66.6 + i^{3.46}) (10 x)}{10 (66.6 + i^{3.46})} = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot - 1$

Step 2.6.1.1.4

Cancel the common factor of $66.6 + i^{3.46}$ .

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Step 2.6.1.1.4.1

Cancel the common factor.

$\frac{(66.6 + i^{3.46}) (10 x)}{10 (66.6 + i^{3.46})} = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot - 1$

Step 2.6.1.1.4.2

Rewrite the expression.

$\frac{10 x}{10} = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot - 1$

Step 2.6.1.1.5

Cancel the common factor of $10$ .

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Step 2.6.1.1.5.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot - 1$

Step 2.6.1.1.5.2

Divide $x$ by $1$ .

$x = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot - 1$

Step 2.6.2

Simplify the right side.

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Step 2.6.2.1

Simplify $\frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10} \cdot - 1$ .

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Step 2.6.2.1.1

Combine $\frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10}$ and $- 1$ .

$x = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46}) \cdot - 1}{10}$

Step 2.6.2.1.2

Simplify the expression.

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Step 2.6.2.1.2.1

Move $- 1$ to the left of $(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})$ .

$x = \frac{- 1 \cdot ((2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46}))}{10}$

Step 2.6.2.1.2.2

Move the negative in front of the fraction.

$x = - \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10}$

Step 2.7

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

$x = \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10}, - \frac{(2 + i^{3.46}) (12 + i^{3.46}) (66.6 + i^{3.46})}{10}$