Trigonometry Examples

$30 + 44 i = 6 + 5 i z$

Step 1

Rewrite the equation as $6 + 5 i z = 30 + 44 i$ .

$6 + 5 i z = 30 + 44 i$

Step 2

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

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Step 2.1

Subtract $6$ from both sides of the equation.

$5 i z = 30 + 44 i - 6$

Step 2.2

Subtract $6$ from $30$ .

$5 i z = 24 + 44 i$

Step 3

Divide each term in $5 i z = 24 + 44 i$ by $5 i$ and simplify.

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Step 3.1

Divide each term in $5 i z = 24 + 44 i$ by $5 i$ .

$\frac{5 i z}{5 i} = \frac{24}{5 i} + \frac{44 i}{5 i}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $5$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{5 i z}{5 i} = \frac{24}{5 i} + \frac{44 i}{5 i}$

Step 3.2.1.2

Rewrite the expression.

$\frac{i z}{i} = \frac{24}{5 i} + \frac{44 i}{5 i}$

Step 3.2.2

Cancel the common factor of $i$ .

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Step 3.2.2.1

Cancel the common factor.

$\frac{i z}{i} = \frac{24}{5 i} + \frac{44 i}{5 i}$

Step 3.2.2.2

Divide $z$ by $1$ .

$z = \frac{24}{5 i} + \frac{44 i}{5 i}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply the numerator and denominator of $\frac{24}{5 i}$ by the conjugate of $5 i$ to make the denominator real.

$z = \frac{24}{5 i} \cdot \frac{i}{i} + \frac{44 i}{5 i}$

Step 3.3.1.2

Multiply.

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Step 3.3.1.2.1

Combine.

$z = \frac{24 i}{5 i i} + \frac{44 i}{5 i}$

Step 3.3.1.2.2

Simplify the denominator.

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Step 3.3.1.2.2.1

Add parentheses.

$z = \frac{24 i}{5 (i i)} + \frac{44 i}{5 i}$

Step 3.3.1.2.2.2

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

$z = \frac{24 i}{5 (i^{1} i)} + \frac{44 i}{5 i}$

Step 3.3.1.2.2.3

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

$z = \frac{24 i}{5 (i^{1} i^{1})} + \frac{44 i}{5 i}$

Step 3.3.1.2.2.4

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

$z = \frac{24 i}{5 i^{1 + 1}} + \frac{44 i}{5 i}$

Step 3.3.1.2.2.5

Add $1$ and $1$ .

$z = \frac{24 i}{5 i^{2}} + \frac{44 i}{5 i}$

Step 3.3.1.2.2.6

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

$z = \frac{24 i}{5 \cdot - 1} + \frac{44 i}{5 i}$

Step 3.3.1.3

Multiply $5$ by $- 1$ .

$z = \frac{24 i}{- 5} + \frac{44 i}{5 i}$

Step 3.3.1.4

Move the negative in front of the fraction.

$z = - \frac{24 i}{5} + \frac{44 i}{5 i}$

Step 3.3.1.5

Cancel the common factor of $i$ .

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Step 3.3.1.5.1

Cancel the common factor.

$z = - \frac{24 i}{5} + \frac{44 i}{5 i}$

Step 3.3.1.5.2

Rewrite the expression.

$z = - \frac{24 i}{5} + \frac{44}{5}$

Step 3.3.2

Reorder $- \frac{24 i}{5}$ and $\frac{44}{5}$ .

$z = \frac{44}{5} - \frac{24 i}{5}$