Trigonometry Examples

${(3 + 4 i)}^{2} - 2 (x - y i) = x + y i$

Step 1

Solve for $y$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

${(3 + 4 i)}^{2} - 2 x - 2 (- y i) = x + y i$

Step 1.1.2

Multiply $- 1$ by $- 2$ .

${(3 + 4 i)}^{2} - 2 x + 2 y i = x + y i$

Step 1.2

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

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Step 1.2.1

Subtract $y i$ from both sides of the equation.

${(3 + 4 i)}^{2} - 2 x + 2 y i - y i = x$

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Rewrite ${(3 + 4 i)}^{2}$ as $(3 + 4 i) (3 + 4 i)$ .

$(3 + 4 i) (3 + 4 i) - 2 x + 2 y i - y i = x$

Step 1.2.2.2

Expand $(3 + 4 i) (3 + 4 i)$ using the FOIL Method.

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Step 1.2.2.2.1

Apply the distributive property.

$3 (3 + 4 i) + 4 i (3 + 4 i) - 2 x + 2 y i - y i = x$

Step 1.2.2.2.2

Apply the distributive property.

$3 \cdot 3 + 3 (4 i) + 4 i (3 + 4 i) - 2 x + 2 y i - y i = x$

Step 1.2.2.2.3

Apply the distributive property.

$3 \cdot 3 + 3 (4 i) + 4 i \cdot 3 + 4 i (4 i) - 2 x + 2 y i - y i = x$

Step 1.2.2.3

Simplify and combine like terms.

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Step 1.2.2.3.1

Simplify each term.

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Step 1.2.2.3.1.1

Multiply $3$ by $3$ .

$9 + 3 (4 i) + 4 i \cdot 3 + 4 i (4 i) - 2 x + 2 y i - y i = x$

Step 1.2.2.3.1.2

Multiply $4$ by $3$ .

$9 + 12 i + 4 i \cdot 3 + 4 i (4 i) - 2 x + 2 y i - y i = x$

Step 1.2.2.3.1.3

Multiply $3$ by $4$ .

$9 + 12 i + 12 i + 4 i (4 i) - 2 x + 2 y i - y i = x$

Step 1.2.2.3.1.4

Multiply $4 i (4 i)$ .

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Step 1.2.2.3.1.4.1

Multiply $4$ by $4$ .

$9 + 12 i + 12 i + 16 i i - 2 x + 2 y i - y i = x$

Step 1.2.2.3.1.4.2

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

$9 + 12 i + 12 i + 16 (i^{1} i) - 2 x + 2 y i - y i = x$

Step 1.2.2.3.1.4.3

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

$9 + 12 i + 12 i + 16 (i^{1} i^{1}) - 2 x + 2 y i - y i = x$

Step 1.2.2.3.1.4.4

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

$9 + 12 i + 12 i + 16 i^{1 + 1} - 2 x + 2 y i - y i = x$

Step 1.2.2.3.1.4.5

Add $1$ and $1$ .

$9 + 12 i + 12 i + 16 i^{2} - 2 x + 2 y i - y i = x$

Step 1.2.2.3.1.5

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

$9 + 12 i + 12 i + 16 \cdot - 1 - 2 x + 2 y i - y i = x$

Step 1.2.2.3.1.6

Multiply $16$ by $- 1$ .

$9 + 12 i + 12 i - 16 - 2 x + 2 y i - y i = x$

Step 1.2.2.3.2

Subtract $16$ from $9$ .

$- 7 + 12 i + 12 i - 2 x + 2 y i - y i = x$

Step 1.2.2.3.3

Add $12 i$ and $12 i$ .

$- 7 + 24 i - 2 x + 2 y i - y i = x$

Step 1.2.3

Subtract $y i$ from $2 y i$ .

$- 7 + 24 i - 2 x + 1 y i = x$

Step 1.2.4

Multiply $y$ by $1$ .

$- 7 + 24 i - 2 x + y i = x$

Step 1.3

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

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Step 1.3.1

Add $7$ to both sides of the equation.

$24 i - 2 x + y i = x + 7$

Step 1.3.2

Subtract $24 i$ from both sides of the equation.

$- 2 x + y i = x + 7 - 24 i$

Step 1.3.3

Add $2 x$ to both sides of the equation.

$y i = x + 7 - 24 i + 2 x$

Step 1.3.4

Add $x$ and $2 x$ .

$y i = 3 x + 7 - 24 i$

Step 1.4

Divide each term in $y i = 3 x + 7 - 24 i$ by $i$ and simplify.

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Step 1.4.1

Divide each term in $y i = 3 x + 7 - 24 i$ by $i$ .

$\frac{y i}{i} = \frac{3 x}{i} + \frac{7}{i} + \frac{- 24 i}{i}$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Cancel the common factor of $i$ .

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Step 1.4.2.1.1

Cancel the common factor.

$\frac{y i}{i} = \frac{3 x}{i} + \frac{7}{i} + \frac{- 24 i}{i}$

Step 1.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{3 x}{i} + \frac{7}{i} + \frac{- 24 i}{i}$

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

Simplify each term.

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Step 1.4.3.1.1

Multiply the numerator and denominator of $\frac{3 x}{i}$ by the conjugate of $i$ to make the denominator real.

$y = \frac{3 x}{i} \cdot \frac{i}{i} + \frac{7}{i} + \frac{- 24 i}{i}$

Step 1.4.3.1.2

Multiply.

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Step 1.4.3.1.2.1

Combine.

$y = \frac{3 x i}{i i} + \frac{7}{i} + \frac{- 24 i}{i}$

Step 1.4.3.1.2.2

Simplify the denominator.

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Step 1.4.3.1.2.2.1

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

$y = \frac{3 x i}{i^{1} i} + \frac{7}{i} + \frac{- 24 i}{i}$

Step 1.4.3.1.2.2.2

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

$y = \frac{3 x i}{i^{1} i^{1}} + \frac{7}{i} + \frac{- 24 i}{i}$

Step 1.4.3.1.2.2.3

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

$y = \frac{3 x i}{i^{1 + 1}} + \frac{7}{i} + \frac{- 24 i}{i}$

Step 1.4.3.1.2.2.4

Add $1$ and $1$ .

$y = \frac{3 x i}{i^{2}} + \frac{7}{i} + \frac{- 24 i}{i}$

Step 1.4.3.1.2.2.5

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

$y = \frac{3 x i}{- 1} + \frac{7}{i} + \frac{- 24 i}{i}$

Step 1.4.3.1.3

Move the negative one from the denominator of $\frac{3 x i}{- 1}$ .

$y = - 1 \cdot (3 x i) + \frac{7}{i} + \frac{- 24 i}{i}$

Step 1.4.3.1.4

Rewrite $- 1 \cdot (3 x i)$ as $- (3 x i)$ .

$y = - (3 x i) + \frac{7}{i} + \frac{- 24 i}{i}$

Step 1.4.3.1.5

Multiply $3$ by $- 1$ .

$y = - 3 (x i) + \frac{7}{i} + \frac{- 24 i}{i}$

Step 1.4.3.1.6

Multiply the numerator and denominator of $\frac{7}{i}$ by the conjugate of $i$ to make the denominator real.

$y = - 3 x i + \frac{7}{i} \cdot \frac{i}{i} + \frac{- 24 i}{i}$

Step 1.4.3.1.7

Multiply.

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Step 1.4.3.1.7.1

Combine.

$y = - 3 x i + \frac{7 i}{i i} + \frac{- 24 i}{i}$

Step 1.4.3.1.7.2

Simplify the denominator.

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Step 1.4.3.1.7.2.1

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

$y = - 3 x i + \frac{7 i}{i^{1} i} + \frac{- 24 i}{i}$

Step 1.4.3.1.7.2.2

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

$y = - 3 x i + \frac{7 i}{i^{1} i^{1}} + \frac{- 24 i}{i}$

Step 1.4.3.1.7.2.3

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

$y = - 3 x i + \frac{7 i}{i^{1 + 1}} + \frac{- 24 i}{i}$

Step 1.4.3.1.7.2.4

Add $1$ and $1$ .

$y = - 3 x i + \frac{7 i}{i^{2}} + \frac{- 24 i}{i}$

Step 1.4.3.1.7.2.5

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

$y = - 3 x i + \frac{7 i}{- 1} + \frac{- 24 i}{i}$

Step 1.4.3.1.8

Move the negative one from the denominator of $\frac{7 i}{- 1}$ .

$y = - 3 x i - 1 \cdot (7 i) + \frac{- 24 i}{i}$

Step 1.4.3.1.9

Rewrite $- 1 \cdot (7 i)$ as $- (7 i)$ .

$y = - 3 x i - (7 i) + \frac{- 24 i}{i}$

Step 1.4.3.1.10

Multiply $7$ by $- 1$ .

$y = - 3 x i - 7 i + \frac{- 24 i}{i}$

Step 1.4.3.1.11

Cancel the common factor of $i$ .

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Step 1.4.3.1.11.1

Cancel the common factor.

$y = - 3 x i - 7 i + \frac{- 24 i}{i}$

Step 1.4.3.1.11.2

Divide $- 24$ by $1$ .

$y = - 3 x i - 7 i - 24$

Step 2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.