Algebra Examples

$\frac{4}{5} (\cos (\frac{π}{8} - \frac{π}{10}) + i \sin (\frac{π}{8} - \frac{π}{10}))$

Step 1

Simplify each term.

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Step 1.1

To write $\frac{π}{8}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{4}{5} (\cos (\frac{π}{8} \cdot \frac{5}{5} - \frac{π}{10}) + i \sin (\frac{π}{8} - \frac{π}{10}))$

Step 1.2

To write $- \frac{π}{10}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{4}{5} (\cos (\frac{π}{8} \cdot \frac{5}{5} - \frac{π}{10} \cdot \frac{4}{4}) + i \sin (\frac{π}{8} - \frac{π}{10}))$

Step 1.3

Write each expression with a common denominator of $40$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{π}{8}$ by $\frac{5}{5}$ .

$\frac{4}{5} (\cos (\frac{π \cdot 5}{8 \cdot 5} - \frac{π}{10} \cdot \frac{4}{4}) + i \sin (\frac{π}{8} - \frac{π}{10}))$

Step 1.3.2

Multiply $8$ by $5$ .

$\frac{4}{5} (\cos (\frac{π \cdot 5}{40} - \frac{π}{10} \cdot \frac{4}{4}) + i \sin (\frac{π}{8} - \frac{π}{10}))$

Step 1.3.3

Multiply $\frac{π}{10}$ by $\frac{4}{4}$ .

$\frac{4}{5} (\cos (\frac{π \cdot 5}{40} - \frac{π \cdot 4}{10 \cdot 4}) + i \sin (\frac{π}{8} - \frac{π}{10}))$

Step 1.3.4

Multiply $10$ by $4$ .

$\frac{4}{5} (\cos (\frac{π \cdot 5}{40} - \frac{π \cdot 4}{40}) + i \sin (\frac{π}{8} - \frac{π}{10}))$

Step 1.4

Combine the numerators over the common denominator.

$\frac{4}{5} (\cos (\frac{π \cdot 5 - π \cdot 4}{40}) + i \sin (\frac{π}{8} - \frac{π}{10}))$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Move $5$ to the left of $π$ .

$\frac{4}{5} (\cos (\frac{5 \cdot π - π \cdot 4}{40}) + i \sin (\frac{π}{8} - \frac{π}{10}))$

Step 1.5.2

Multiply $4$ by $- 1$ .

$\frac{4}{5} (\cos (\frac{5 π - 4 π}{40}) + i \sin (\frac{π}{8} - \frac{π}{10}))$

Step 1.5.3

Subtract $4 π$ from $5 π$ .

$\frac{4}{5} (\cos (\frac{π}{40}) + i \sin (\frac{π}{8} - \frac{π}{10}))$

Step 1.6

To write $\frac{π}{8}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{4}{5} (\cos (\frac{π}{40}) + i \sin (\frac{π}{8} \cdot \frac{5}{5} - \frac{π}{10}))$

Step 1.7

To write $- \frac{π}{10}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{4}{5} (\cos (\frac{π}{40}) + i \sin (\frac{π}{8} \cdot \frac{5}{5} - \frac{π}{10} \cdot \frac{4}{4}))$

Step 1.8

Write each expression with a common denominator of $40$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.8.1

Multiply $\frac{π}{8}$ by $\frac{5}{5}$ .

$\frac{4}{5} (\cos (\frac{π}{40}) + i \sin (\frac{π \cdot 5}{8 \cdot 5} - \frac{π}{10} \cdot \frac{4}{4}))$

Step 1.8.2

Multiply $8$ by $5$ .

$\frac{4}{5} (\cos (\frac{π}{40}) + i \sin (\frac{π \cdot 5}{40} - \frac{π}{10} \cdot \frac{4}{4}))$

Step 1.8.3

Multiply $\frac{π}{10}$ by $\frac{4}{4}$ .

$\frac{4}{5} (\cos (\frac{π}{40}) + i \sin (\frac{π \cdot 5}{40} - \frac{π \cdot 4}{10 \cdot 4}))$

Step 1.8.4

Multiply $10$ by $4$ .

$\frac{4}{5} (\cos (\frac{π}{40}) + i \sin (\frac{π \cdot 5}{40} - \frac{π \cdot 4}{40}))$

Step 1.9

Combine the numerators over the common denominator.

$\frac{4}{5} (\cos (\frac{π}{40}) + i \sin (\frac{π \cdot 5 - π \cdot 4}{40}))$

Step 1.10

Simplify the numerator.

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Step 1.10.1

Move $5$ to the left of $π$ .

$\frac{4}{5} (\cos (\frac{π}{40}) + i \sin (\frac{5 \cdot π - π \cdot 4}{40}))$

Step 1.10.2

Multiply $4$ by $- 1$ .

$\frac{4}{5} (\cos (\frac{π}{40}) + i \sin (\frac{5 π - 4 π}{40}))$

Step 1.10.3

Subtract $4 π$ from $5 π$ .

$\frac{4}{5} (\cos (\frac{π}{40}) + i \sin (\frac{π}{40}))$

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

$\frac{4}{5} \cos (\frac{π}{40}) + \frac{4}{5} (i \sin (\frac{π}{40}))$

Step 2.2

Combine $\frac{4}{5}$ and $\cos (\frac{π}{40})$ .

$\frac{4 \cos (\frac{π}{40})}{5} + \frac{4}{5} (i \sin (\frac{π}{40}))$

Step 3

Multiply $\frac{4}{5} (i \sin (\frac{π}{40}))$ .

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Step 3.1

Combine $i$ and $\frac{4}{5}$ .

$\frac{4 \cos (\frac{π}{40})}{5} + \frac{i \cdot 4}{5} \sin (\frac{π}{40})$

Step 3.2

Combine $\frac{i \cdot 4}{5}$ and $\sin (\frac{π}{40})$ .

$\frac{4 \cos (\frac{π}{40})}{5} + \frac{i \cdot 4 \sin (\frac{π}{40})}{5}$

Step 4

Move $4$ to the left of $i$ .

$\frac{4 \cos (\frac{π}{40})}{5} + \frac{4 i \sin (\frac{π}{40})}{5}$