Trigonometry Examples

$\frac{i}{3 - 2 i} - \frac{2 i}{3 + 8 i}$

Step 1

Simplify each term.

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Step 1.1

Multiply the numerator and denominator of $\frac{i}{3 - 2 i}$ by the conjugate of $3 - 2 i$ to make the denominator real.

$\frac{i}{3 - 2 i} \cdot \frac{3 + 2 i}{3 + 2 i} - \frac{2 i}{3 + 8 i}$

Step 1.2

Multiply.

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Step 1.2.1

Combine.

$\frac{i (3 + 2 i)}{(3 - 2 i) (3 + 2 i)} - \frac{2 i}{3 + 8 i}$

Step 1.2.2

Simplify the numerator.

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Step 1.2.2.1

Apply the distributive property.

$\frac{i \cdot 3 + i (2 i)}{(3 - 2 i) (3 + 2 i)} - \frac{2 i}{3 + 8 i}$

Step 1.2.2.2

Move $3$ to the left of $i$ .

$\frac{3 \cdot i + i (2 i)}{(3 - 2 i) (3 + 2 i)} - \frac{2 i}{3 + 8 i}$

Step 1.2.2.3

Multiply $i (2 i)$ .

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Step 1.2.2.3.1

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

$\frac{3 \cdot i + 2 (i^{1} i)}{(3 - 2 i) (3 + 2 i)} - \frac{2 i}{3 + 8 i}$

Step 1.2.2.3.2

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

$\frac{3 \cdot i + 2 (i^{1} i^{1})}{(3 - 2 i) (3 + 2 i)} - \frac{2 i}{3 + 8 i}$

Step 1.2.2.3.3

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

$\frac{3 \cdot i + 2 i^{1 + 1}}{(3 - 2 i) (3 + 2 i)} - \frac{2 i}{3 + 8 i}$

Step 1.2.2.3.4

Add $1$ and $1$ .

$\frac{3 \cdot i + 2 i^{2}}{(3 - 2 i) (3 + 2 i)} - \frac{2 i}{3 + 8 i}$

Step 1.2.2.4

Simplify each term.

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Step 1.2.2.4.1

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

$\frac{3 i + 2 \cdot - 1}{(3 - 2 i) (3 + 2 i)} - \frac{2 i}{3 + 8 i}$

Step 1.2.2.4.2

Multiply $2$ by $- 1$ .

$\frac{3 i - 2}{(3 - 2 i) (3 + 2 i)} - \frac{2 i}{3 + 8 i}$

Step 1.2.2.5

Reorder $3 i$ and $- 2$ .

$\frac{- 2 + 3 i}{(3 - 2 i) (3 + 2 i)} - \frac{2 i}{3 + 8 i}$

Step 1.2.3

Simplify the denominator.

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Step 1.2.3.1

Expand $(3 - 2 i) (3 + 2 i)$ using the FOIL Method.

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Step 1.2.3.1.1

Apply the distributive property.

$\frac{- 2 + 3 i}{3 (3 + 2 i) - 2 i (3 + 2 i)} - \frac{2 i}{3 + 8 i}$

Step 1.2.3.1.2

Apply the distributive property.

$\frac{- 2 + 3 i}{3 \cdot 3 + 3 (2 i) - 2 i (3 + 2 i)} - \frac{2 i}{3 + 8 i}$

Step 1.2.3.1.3

Apply the distributive property.

$\frac{- 2 + 3 i}{3 \cdot 3 + 3 (2 i) - 2 i \cdot 3 - 2 i (2 i)} - \frac{2 i}{3 + 8 i}$

Step 1.2.3.2

Simplify.

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Step 1.2.3.2.1

Multiply $3$ by $3$ .

$\frac{- 2 + 3 i}{9 + 3 (2 i) - 2 i \cdot 3 - 2 i (2 i)} - \frac{2 i}{3 + 8 i}$

Step 1.2.3.2.2

Multiply $2$ by $3$ .

$\frac{- 2 + 3 i}{9 + 6 i - 2 i \cdot 3 - 2 i (2 i)} - \frac{2 i}{3 + 8 i}$

Step 1.2.3.2.3

Multiply $3$ by $- 2$ .

$\frac{- 2 + 3 i}{9 + 6 i - 6 i - 2 i (2 i)} - \frac{2 i}{3 + 8 i}$

Step 1.2.3.2.4

Multiply $2$ by $- 2$ .

$\frac{- 2 + 3 i}{9 + 6 i - 6 i - 4 i i} - \frac{2 i}{3 + 8 i}$

Step 1.2.3.2.5

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

$\frac{- 2 + 3 i}{9 + 6 i - 6 i - 4 (i^{1} i)} - \frac{2 i}{3 + 8 i}$

Step 1.2.3.2.6

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

$\frac{- 2 + 3 i}{9 + 6 i - 6 i - 4 (i^{1} i^{1})} - \frac{2 i}{3 + 8 i}$

Step 1.2.3.2.7

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

$\frac{- 2 + 3 i}{9 + 6 i - 6 i - 4 i^{1 + 1}} - \frac{2 i}{3 + 8 i}$

Step 1.2.3.2.8

Add $1$ and $1$ .

$\frac{- 2 + 3 i}{9 + 6 i - 6 i - 4 i^{2}} - \frac{2 i}{3 + 8 i}$

Step 1.2.3.2.9

Subtract $6 i$ from $6 i$ .

$\frac{- 2 + 3 i}{9 + 0 - 4 i^{2}} - \frac{2 i}{3 + 8 i}$

Step 1.2.3.2.10

Add $9$ and $0$ .

$\frac{- 2 + 3 i}{9 - 4 i^{2}} - \frac{2 i}{3 + 8 i}$

Step 1.2.3.3

Simplify each term.

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Step 1.2.3.3.1

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

$\frac{- 2 + 3 i}{9 - 4 \cdot - 1} - \frac{2 i}{3 + 8 i}$

Step 1.2.3.3.2

Multiply $- 4$ by $- 1$ .

$\frac{- 2 + 3 i}{9 + 4} - \frac{2 i}{3 + 8 i}$

Step 1.2.3.4

Add $9$ and $4$ .

$\frac{- 2 + 3 i}{13} - \frac{2 i}{3 + 8 i}$

Step 1.3

Split the fraction $\frac{- 2 + 3 i}{13}$ into two fractions.

$\frac{- 2}{13} + \frac{3 i}{13} - \frac{2 i}{3 + 8 i}$

Step 1.4

Move the negative in front of the fraction.

$- \frac{2}{13} + \frac{3 i}{13} - \frac{2 i}{3 + 8 i}$

Step 1.5

Multiply the numerator and denominator of $\frac{2 i}{3 + 8 i}$ by the conjugate of $3 + 8 i$ to make the denominator real.

$- \frac{2}{13} + \frac{3 i}{13} - (\frac{2 i}{3 + 8 i} \cdot \frac{3 - 8 i}{3 - 8 i})$

Step 1.6

Multiply.

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Step 1.6.1

Combine.

$- \frac{2}{13} + \frac{3 i}{13} - \frac{2 i (3 - 8 i)}{(3 + 8 i) (3 - 8 i)}$

Step 1.6.2

Simplify the numerator.

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Step 1.6.2.1

Apply the distributive property.

$- \frac{2}{13} + \frac{3 i}{13} - \frac{2 i \cdot 3 + 2 i (- 8 i)}{(3 + 8 i) (3 - 8 i)}$

Step 1.6.2.2

Multiply $3$ by $2$ .

$- \frac{2}{13} + \frac{3 i}{13} - \frac{6 i + 2 i (- 8 i)}{(3 + 8 i) (3 - 8 i)}$

Step 1.6.2.3

Multiply $2 i (- 8 i)$ .

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Step 1.6.2.3.1

Multiply $- 8$ by $2$ .

$- \frac{2}{13} + \frac{3 i}{13} - \frac{6 i - 16 i i}{(3 + 8 i) (3 - 8 i)}$

Step 1.6.2.3.2

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

$- \frac{2}{13} + \frac{3 i}{13} - \frac{6 i - 16 (i^{1} i)}{(3 + 8 i) (3 - 8 i)}$

Step 1.6.2.3.3

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

$- \frac{2}{13} + \frac{3 i}{13} - \frac{6 i - 16 (i^{1} i^{1})}{(3 + 8 i) (3 - 8 i)}$

Step 1.6.2.3.4

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

$- \frac{2}{13} + \frac{3 i}{13} - \frac{6 i - 16 i^{1 + 1}}{(3 + 8 i) (3 - 8 i)}$

Step 1.6.2.3.5

Add $1$ and $1$ .

$- \frac{2}{13} + \frac{3 i}{13} - \frac{6 i - 16 i^{2}}{(3 + 8 i) (3 - 8 i)}$

Step 1.6.2.4

Simplify each term.

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Step 1.6.2.4.1

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

$- \frac{2}{13} + \frac{3 i}{13} - \frac{6 i - 16 \cdot - 1}{(3 + 8 i) (3 - 8 i)}$

Step 1.6.2.4.2

Multiply $- 16$ by $- 1$ .

$- \frac{2}{13} + \frac{3 i}{13} - \frac{6 i + 16}{(3 + 8 i) (3 - 8 i)}$

Step 1.6.2.5

Reorder $6 i$ and $16$ .

$- \frac{2}{13} + \frac{3 i}{13} - \frac{16 + 6 i}{(3 + 8 i) (3 - 8 i)}$

Step 1.6.3

Simplify the denominator.

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Step 1.6.3.1

Expand $(3 + 8 i) (3 - 8 i)$ using the FOIL Method.

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Step 1.6.3.1.1

Apply the distributive property.

$- \frac{2}{13} + \frac{3 i}{13} - \frac{16 + 6 i}{3 (3 - 8 i) + 8 i (3 - 8 i)}$

Step 1.6.3.1.2

Apply the distributive property.

$- \frac{2}{13} + \frac{3 i}{13} - \frac{16 + 6 i}{3 \cdot 3 + 3 (- 8 i) + 8 i (3 - 8 i)}$

Step 1.6.3.1.3

Apply the distributive property.

$- \frac{2}{13} + \frac{3 i}{13} - \frac{16 + 6 i}{3 \cdot 3 + 3 (- 8 i) + 8 i \cdot 3 + 8 i (- 8 i)}$

Step 1.6.3.2

Simplify.

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Step 1.6.3.2.1

Multiply $3$ by $3$ .

$- \frac{2}{13} + \frac{3 i}{13} - \frac{16 + 6 i}{9 + 3 (- 8 i) + 8 i \cdot 3 + 8 i (- 8 i)}$

Step 1.6.3.2.2

Multiply $- 8$ by $3$ .

$- \frac{2}{13} + \frac{3 i}{13} - \frac{16 + 6 i}{9 - 24 i + 8 i \cdot 3 + 8 i (- 8 i)}$

Step 1.6.3.2.3

Multiply $3$ by $8$ .

$- \frac{2}{13} + \frac{3 i}{13} - \frac{16 + 6 i}{9 - 24 i + 24 i + 8 i (- 8 i)}$

Step 1.6.3.2.4

Multiply $- 8$ by $8$ .

$- \frac{2}{13} + \frac{3 i}{13} - \frac{16 + 6 i}{9 - 24 i + 24 i - 64 i i}$

Step 1.6.3.2.5

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

$- \frac{2}{13} + \frac{3 i}{13} - \frac{16 + 6 i}{9 - 24 i + 24 i - 64 (i^{1} i)}$

Step 1.6.3.2.6

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

$- \frac{2}{13} + \frac{3 i}{13} - \frac{16 + 6 i}{9 - 24 i + 24 i - 64 (i^{1} i^{1})}$

Step 1.6.3.2.7

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

$- \frac{2}{13} + \frac{3 i}{13} - \frac{16 + 6 i}{9 - 24 i + 24 i - 64 i^{1 + 1}}$

Step 1.6.3.2.8

Add $1$ and $1$ .

$- \frac{2}{13} + \frac{3 i}{13} - \frac{16 + 6 i}{9 - 24 i + 24 i - 64 i^{2}}$

Step 1.6.3.2.9

Add $- 24 i$ and $24 i$ .

$- \frac{2}{13} + \frac{3 i}{13} - \frac{16 + 6 i}{9 + 0 - 64 i^{2}}$

Step 1.6.3.2.10

Add $9$ and $0$ .

$- \frac{2}{13} + \frac{3 i}{13} - \frac{16 + 6 i}{9 - 64 i^{2}}$

Step 1.6.3.3

Simplify each term.

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Step 1.6.3.3.1

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

$- \frac{2}{13} + \frac{3 i}{13} - \frac{16 + 6 i}{9 - 64 \cdot - 1}$

Step 1.6.3.3.2

Multiply $- 64$ by $- 1$ .

$- \frac{2}{13} + \frac{3 i}{13} - \frac{16 + 6 i}{9 + 64}$

Step 1.6.3.4

Add $9$ and $64$ .

$- \frac{2}{13} + \frac{3 i}{13} - \frac{16 + 6 i}{73}$

Step 1.7

Split the fraction $\frac{16 + 6 i}{73}$ into two fractions.

$- \frac{2}{13} + \frac{3 i}{13} - (\frac{16}{73} + \frac{6 i}{73})$

Step 1.8

Apply the distributive property.

$- \frac{2}{13} + \frac{3 i}{13} - \frac{16}{73} - \frac{6 i}{73}$

Step 2

To write $- \frac{2}{13}$ as a fraction with a common denominator, multiply by $\frac{73}{73}$ .

$- \frac{2}{13} \cdot \frac{73}{73} - \frac{16}{73} + \frac{3 i}{13} - \frac{6 i}{73}$

Step 3

To write $- \frac{16}{73}$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$- \frac{2}{13} \cdot \frac{73}{73} - \frac{16}{73} \cdot \frac{13}{13} + \frac{3 i}{13} - \frac{6 i}{73}$

Step 4

Write each expression with a common denominator of $949$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2}{13}$ by $\frac{73}{73}$ .

$- \frac{2 \cdot 73}{13 \cdot 73} - \frac{16}{73} \cdot \frac{13}{13} + \frac{3 i}{13} - \frac{6 i}{73}$

Step 4.2

Multiply $13$ by $73$ .

$- \frac{2 \cdot 73}{949} - \frac{16}{73} \cdot \frac{13}{13} + \frac{3 i}{13} - \frac{6 i}{73}$

Step 4.3

Multiply $\frac{16}{73}$ by $\frac{13}{13}$ .

$- \frac{2 \cdot 73}{949} - \frac{16 \cdot 13}{73 \cdot 13} + \frac{3 i}{13} - \frac{6 i}{73}$

Step 4.4

Multiply $73$ by $13$ .

$- \frac{2 \cdot 73}{949} - \frac{16 \cdot 13}{949} + \frac{3 i}{13} - \frac{6 i}{73}$

Step 5

Combine the numerators over the common denominator.

$\frac{- 2 \cdot 73 - 16 \cdot 13}{949} + \frac{3 i}{13} - \frac{6 i}{73}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 2$ by $73$ .

$\frac{- 146 - 16 \cdot 13}{949} + \frac{3 i}{13} - \frac{6 i}{73}$

Step 6.2

Multiply $- 16$ by $13$ .

$\frac{- 146 - 208}{949} + \frac{3 i}{13} - \frac{6 i}{73}$

Step 6.3

Subtract $208$ from $- 146$ .

$\frac{- 354}{949} + \frac{3 i}{13} - \frac{6 i}{73}$

Step 7

Move the negative in front of the fraction.

$- \frac{354}{949} + \frac{3 i}{13} - \frac{6 i}{73}$

Step 8

To write $\frac{3 i}{13}$ as a fraction with a common denominator, multiply by $\frac{73}{73}$ .

$- \frac{354}{949} + \frac{3 i}{13} \cdot \frac{73}{73} - \frac{6 i}{73}$

Step 9

To write $- \frac{6 i}{73}$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$- \frac{354}{949} + \frac{3 i}{13} \cdot \frac{73}{73} - \frac{6 i}{73} \cdot \frac{13}{13}$

Step 10

Write each expression with a common denominator of $949$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.1

Multiply $\frac{3 i}{13}$ by $\frac{73}{73}$ .

$- \frac{354}{949} + \frac{3 i \cdot 73}{13 \cdot 73} - \frac{6 i}{73} \cdot \frac{13}{13}$

Step 10.2

Multiply $13$ by $73$ .

$- \frac{354}{949} + \frac{3 i \cdot 73}{949} - \frac{6 i}{73} \cdot \frac{13}{13}$

Step 10.3

Multiply $\frac{6 i}{73}$ by $\frac{13}{13}$ .

$- \frac{354}{949} + \frac{3 i \cdot 73}{949} - \frac{6 i \cdot 13}{73 \cdot 13}$

Step 10.4

Multiply $73$ by $13$ .

$- \frac{354}{949} + \frac{3 i \cdot 73}{949} - \frac{6 i \cdot 13}{949}$

Step 11

Combine the numerators over the common denominator.

$- \frac{354}{949} + \frac{141 i}{949}$