Trigonometry Examples

$3 + x i = (x + 1) + 2 i$

Step 1

Subtract $x$ from both sides of the equation.

$3 + x i - x = 1 + 2 i$

Step 2

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

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Step 2.1

Subtract $3$ from both sides of the equation.

$x i - x = 1 + 2 i - 3$

Step 2.2

Subtract $3$ from $1$ .

$x i - x = - 2 + 2 i$

Step 3

Factor $x$ out of $x i - x$ .

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Step 3.1

Factor $x$ out of $x i$ .

$x (i) - x = - 2 + 2 i$

Step 3.2

Factor $x$ out of $- x$ .

$x (i) + x \cdot - 1 = - 2 + 2 i$

Step 3.3

Factor $x$ out of $x (i) + x \cdot - 1$ .

$x (i - 1) = - 2 + 2 i$

Step 4

Divide each term in $x (i - 1) = - 2 + 2 i$ by $i - 1$ and simplify.

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Step 4.1

Divide each term in $x (i - 1) = - 2 + 2 i$ by $i - 1$ .

$\frac{x (i - 1)}{i - 1} = \frac{- 2}{i - 1} + \frac{2 i}{i - 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $i - 1$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{x (i - 1)}{i - 1} = \frac{- 2}{i - 1} + \frac{2 i}{i - 1}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{i - 1} + \frac{2 i}{i - 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Combine the numerators over the common denominator.

$x = \frac{- 2 + 2 i}{i - 1}$

Step 4.3.2

Multiply the numerator and denominator of $\frac{- 2 + 2 i}{- 1 + 1 i}$ by the conjugate of $- 1 + 1 i$ to make the denominator real.

$x = \frac{- 2 + 2 i}{- 1 + 1 i} \cdot \frac{- 1 - i}{- 1 - i}$

Step 4.3.3

Multiply.

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Step 4.3.3.1

Combine.

$x = \frac{(- 2 + 2 i) (- 1 - i)}{(- 1 + 1 i) (- 1 - i)}$

Step 4.3.3.2

Simplify the numerator.

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Step 4.3.3.2.1

Expand $(- 2 + 2 i) (- 1 - i)$ using the FOIL Method.

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Step 4.3.3.2.1.1

Apply the distributive property.

$x = \frac{- 2 (- 1 - i) + 2 i (- 1 - i)}{(- 1 + 1 i) (- 1 - i)}$

Step 4.3.3.2.1.2

Apply the distributive property.

$x = \frac{- 2 \cdot - 1 - 2 (- i) + 2 i (- 1 - i)}{(- 1 + 1 i) (- 1 - i)}$

Step 4.3.3.2.1.3

Apply the distributive property.

$x = \frac{- 2 \cdot - 1 - 2 (- i) + 2 i \cdot - 1 + 2 i (- i)}{(- 1 + 1 i) (- 1 - i)}$

Step 4.3.3.2.2

Simplify and combine like terms.

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Step 4.3.3.2.2.1

Simplify each term.

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Step 4.3.3.2.2.1.1

Multiply $- 2$ by $- 1$ .

$x = \frac{2 - 2 (- i) + 2 i \cdot - 1 + 2 i (- i)}{(- 1 + 1 i) (- 1 - i)}$

Step 4.3.3.2.2.1.2

Multiply $- 1$ by $- 2$ .

$x = \frac{2 + 2 i + 2 i \cdot - 1 + 2 i (- i)}{(- 1 + 1 i) (- 1 - i)}$

Step 4.3.3.2.2.1.3

Multiply $- 1$ by $2$ .

$x = \frac{2 + 2 i - 2 i + 2 i (- i)}{(- 1 + 1 i) (- 1 - i)}$

Step 4.3.3.2.2.1.4

Multiply $2 i (- i)$ .

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Step 4.3.3.2.2.1.4.1

Multiply $- 1$ by $2$ .

$x = \frac{2 + 2 i - 2 i - 2 i i}{(- 1 + 1 i) (- 1 - i)}$

Step 4.3.3.2.2.1.4.2

Raise $i$ to the power of $1$ .

$x = \frac{2 + 2 i - 2 i - 2 (i^{1} i)}{(- 1 + 1 i) (- 1 - i)}$

Step 4.3.3.2.2.1.4.3

Raise $i$ to the power of $1$ .

$x = \frac{2 + 2 i - 2 i - 2 (i^{1} i^{1})}{(- 1 + 1 i) (- 1 - i)}$

Step 4.3.3.2.2.1.4.4

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

$x = \frac{2 + 2 i - 2 i - 2 i^{1 + 1}}{(- 1 + 1 i) (- 1 - i)}$

Step 4.3.3.2.2.1.4.5

Add $1$ and $1$ .

$x = \frac{2 + 2 i - 2 i - 2 i^{2}}{(- 1 + 1 i) (- 1 - i)}$

Step 4.3.3.2.2.1.5

Rewrite $i^{2}$ as $- 1$ .

$x = \frac{2 + 2 i - 2 i - 2 \cdot - 1}{(- 1 + 1 i) (- 1 - i)}$

Step 4.3.3.2.2.1.6

Multiply $- 2$ by $- 1$ .

$x = \frac{2 + 2 i - 2 i + 2}{(- 1 + 1 i) (- 1 - i)}$

Step 4.3.3.2.2.2

Add $2$ and $2$ .

$x = \frac{4 + 2 i - 2 i}{(- 1 + 1 i) (- 1 - i)}$

Step 4.3.3.2.2.3

Subtract $2 i$ from $2 i$ .

$x = \frac{4 + 0}{(- 1 + 1 i) (- 1 - i)}$

Step 4.3.3.2.2.4

Add $4$ and $0$ .

$x = \frac{4}{(- 1 + 1 i) (- 1 - i)}$

Step 4.3.3.3

Simplify the denominator.

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Step 4.3.3.3.1

Expand $(- 1 + 1 i) (- 1 - i)$ using the FOIL Method.

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Step 4.3.3.3.1.1

Apply the distributive property.

$x = \frac{4}{- 1 (- 1 - i) + 1 i (- 1 - i)}$

Step 4.3.3.3.1.2

Apply the distributive property.

$x = \frac{4}{- 1 \cdot - 1 - 1 (- i) + 1 i (- 1 - i)}$

Step 4.3.3.3.1.3

Apply the distributive property.

$x = \frac{4}{- 1 \cdot - 1 - 1 (- i) + 1 i \cdot - 1 + 1 i (- i)}$

Step 4.3.3.3.2

Simplify.

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Step 4.3.3.3.2.1

Multiply $- 1$ by $- 1$ .

$x = \frac{4}{1 - 1 (- i) + 1 i \cdot - 1 + 1 i (- i)}$

Step 4.3.3.3.2.2

Multiply $- 1$ by $- 1$ .

$x = \frac{4}{1 + 1 i + 1 i \cdot - 1 + 1 i (- i)}$

Step 4.3.3.3.2.3

Multiply $- 1$ by $1$ .

$x = \frac{4}{1 + 1 i - i + 1 i (- i)}$

Step 4.3.3.3.2.4

Multiply $- 1$ by $1$ .

$x = \frac{4}{1 + 1 i - i - i i}$

Step 4.3.3.3.2.5

Raise $i$ to the power of $1$ .

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

Step 4.3.3.3.2.6

Raise $i$ to the power of $1$ .

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

Step 4.3.3.3.2.7

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

$x = \frac{4}{1 + 1 i - i - i^{1 + 1}}$

Step 4.3.3.3.2.8

Add $1$ and $1$ .

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

Step 4.3.3.3.2.9

Subtract $i$ from $1 i$ .

$x = \frac{4}{1 + 0 - i^{2}}$

Step 4.3.3.3.2.10

Add $1$ and $0$ .

$x = \frac{4}{1 - i^{2}}$

Step 4.3.3.3.3

Simplify each term.

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Step 4.3.3.3.3.1

Rewrite $i^{2}$ as $- 1$ .

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

Step 4.3.3.3.3.2

Multiply $- 1$ by $- 1$ .

$x = \frac{4}{1 + 1}$

Step 4.3.3.3.4

Add $1$ and $1$ .

$x = \frac{4}{2}$

Step 4.3.4

Divide $4$ by $2$ .

$x = 2$