Trigonometry Examples

$\frac{z - i}{z - 2} = \frac{3 i + 1}{2}$

Step 1

Subtract $\frac{3 i + 1}{2}$ from both sides of the equation.

$\frac{z - i}{z - 2} - \frac{3 i + 1}{2} = 0$

Step 2

Simplify $\frac{z - i}{z - 2} - \frac{3 i + 1}{2}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Split the fraction $\frac{3 i + 1}{2}$ into two fractions.

$\frac{z - i}{z - 2} - (\frac{3 i}{2} + \frac{1}{2}) = 0$

Step 2.1.2

Apply the distributive property.

$\frac{z - i}{z - 2} - \frac{3 i}{2} - \frac{1}{2} = 0$

Step 2.2

To write $\frac{z - i}{z - 2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{z - i}{z - 2} \cdot \frac{2}{2} - \frac{3 i}{2} - \frac{1}{2} = 0$

Step 2.3

To write $- \frac{3 i}{2}$ as a fraction with a common denominator, multiply by $\frac{z - 2}{z - 2}$ .

$\frac{z - i}{z - 2} \cdot \frac{2}{2} - \frac{3 i}{2} \cdot \frac{z - 2}{z - 2} - \frac{1}{2} = 0$

Step 2.4

Write each expression with a common denominator of $(z - 2) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{z - i}{z - 2}$ by $\frac{2}{2}$ .

$\frac{(z - i) \cdot 2}{(z - 2) \cdot 2} - \frac{3 i}{2} \cdot \frac{z - 2}{z - 2} - \frac{1}{2} = 0$

Step 2.4.2

Multiply $\frac{3 i}{2}$ by $\frac{z - 2}{z - 2}$ .

$\frac{(z - i) \cdot 2}{(z - 2) \cdot 2} - \frac{3 i (z - 2)}{2 (z - 2)} - \frac{1}{2} = 0$

Step 2.4.3

Reorder the factors of $(z - 2) \cdot 2$ .

$\frac{(z - i) \cdot 2}{2 (z - 2)} - \frac{3 i (z - 2)}{2 (z - 2)} - \frac{1}{2} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{(z - i) \cdot 2 - 3 i (z - 2)}{2 (z - 2)} - \frac{1}{2} = 0$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Apply the distributive property.

$\frac{z \cdot 2 - i \cdot 2 - 3 i (z - 2)}{2 (z - 2)} - \frac{1}{2} = 0$

Step 2.6.2

Move $2$ to the left of $z$ .

$\frac{2 \cdot z - i \cdot 2 - 3 i (z - 2)}{2 (z - 2)} - \frac{1}{2} = 0$

Step 2.6.3

Multiply $2$ by $- 1$ .

$\frac{2 \cdot z - 2 i - 3 i (z - 2)}{2 (z - 2)} - \frac{1}{2} = 0$

Step 2.6.4

Apply the distributive property.

$\frac{2 z - 2 i - 3 i z - 3 i \cdot - 2}{2 (z - 2)} - \frac{1}{2} = 0$

Step 2.6.5

Multiply $- 2$ by $- 3$ .

$\frac{2 z - 2 i - 3 i z + 6 i}{2 (z - 2)} - \frac{1}{2} = 0$

Step 2.6.6

Add $- 2 i$ and $6 i$ .

$\frac{2 z - 3 i z + 4 i}{2 (z - 2)} - \frac{1}{2} = 0$

Step 2.6.7

Reorder terms.

$\frac{- 3 i z + 2 z + 4 i}{2 (z - 2)} - \frac{1}{2} = 0$

Step 2.7

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{z - 2}{z - 2}$ .

$\frac{- 3 i z + 2 z + 4 i}{2 (z - 2)} - \frac{1}{2} \cdot \frac{z - 2}{z - 2} = 0$

Step 2.8

Multiply $\frac{1}{2}$ by $\frac{z - 2}{z - 2}$ .

$\frac{- 3 i z + 2 z + 4 i}{2 (z - 2)} - \frac{z - 2}{2 (z - 2)} = 0$

Step 2.9

Combine the numerators over the common denominator.

$\frac{- 3 i z + 2 z + 4 i - (z - 2)}{2 (z - 2)} = 0$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Apply the distributive property.

$\frac{- 3 i z + 2 z + 4 i - z - - 2}{2 (z - 2)} = 0$

Step 2.10.2

Multiply $- 1$ by $- 2$ .

$\frac{- 3 i z + 2 z + 4 i - z + 2}{2 (z - 2)} = 0$

Step 2.10.3

Subtract $z$ from $2 z$ .

$\frac{- 3 i z + z + 4 i + 2}{2 (z - 2)} = 0$

Step 2.10.4

Reorder terms.

$\frac{- 3 i z + z + 2 + 4 i}{2 (z - 2)} = 0$

Step 2.11

Factor $- 1$ out of $- 3 i z$ .

$\frac{- (3 i z) + z + 2 + 4 i}{2 (z - 2)} = 0$

Step 2.12

Factor $- 1$ out of $z$ .

$\frac{- (3 i z) - 1 (- z) + 2 + 4 i}{2 (z - 2)} = 0$

Step 2.13

Factor $- 1$ out of $- (3 i z) - 1 (- z)$ .

$\frac{- (3 i z - z) + 2 + 4 i}{2 (z - 2)} = 0$

Step 2.14

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{- (3 i z - z) - 1 (- 2) + 4 i}{2 (z - 2)} = 0$

Step 2.15

Factor $- 1$ out of $- (3 i z - z) - 1 (- 2)$ .

$\frac{- (3 i z - z - 2) + 4 i}{2 (z - 2)} = 0$

Step 2.16

Factor $- 1$ out of $4 i$ .

$\frac{- (3 i z - z - 2) - (- 4 i)}{2 (z - 2)} = 0$

Step 2.17

Factor $- 1$ out of $- (3 i z - z - 2) - (- 4 i)$ .

$\frac{- (3 i z - z - 2 - 4 i)}{2 (z - 2)} = 0$

Step 2.18

Rewrite $- (3 i z - z - 2 - 4 i)$ as $- 1 (3 i z - z - 2 - 4 i)$ .

$\frac{- 1 (3 i z - z - 2 - 4 i)}{2 (z - 2)} = 0$

Step 2.19

Move the negative in front of the fraction.

$- \frac{3 i z - z - 2 - 4 i}{2 (z - 2)} = 0$

Step 3

Set the denominator in $\frac{3 i z - z - 2 - 4 i}{2 (z - 2)}$ equal to $0$ to find where the expression is undefined.

$2 (z - 2) = 0$

Step 4

Solve for $z$ .

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Step 4.1

Divide each term in $2 (z - 2) = 0$ by $2$ and simplify.

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Step 4.1.1

Divide each term in $2 (z - 2) = 0$ by $2$ .

$\frac{2 (z - 2)}{2} = \frac{0}{2}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $2$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{2 (z - 2)}{2} = \frac{0}{2}$

Step 4.1.2.1.2

Divide $z - 2$ by $1$ .

$z - 2 = \frac{0}{2}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $2$ .

$z - 2 = 0$

Step 4.2

Add $2$ to both sides of the equation.

$z = 2$

Step 5