Trigonometry Examples

$\frac{\sqrt{5} + i \sqrt{15}}{5 \sqrt{15} + 5 i \sqrt{5}}$

Step 1

Multiply $\frac{\sqrt{5} + i \sqrt{15}}{5 \sqrt{15} + 5 i \sqrt{5}}$ by $\frac{5 \sqrt{15} - 5 i \sqrt{5}}{5 \sqrt{15} - 5 i \sqrt{5}}$ .

$\frac{\sqrt{5} + i \sqrt{15}}{5 \sqrt{15} + 5 i \sqrt{5}} \cdot \frac{5 \sqrt{15} - 5 i \sqrt{5}}{5 \sqrt{15} - 5 i \sqrt{5}}$

Step 2

Simplify terms.

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Step 2.1

Multiply $\frac{\sqrt{5} + i \sqrt{15}}{5 \sqrt{15} + 5 i \sqrt{5}}$ by $\frac{5 \sqrt{15} - 5 i \sqrt{5}}{5 \sqrt{15} - 5 i \sqrt{5}}$ .

$\frac{(\sqrt{5} + i \sqrt{15}) (5 \sqrt{15} - 5 i \sqrt{5})}{(5 \sqrt{15} + 5 i \sqrt{5}) (5 \sqrt{15} - 5 i \sqrt{5})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{5} + i \sqrt{15}) (5 \sqrt{15} - 5 i \sqrt{5})}{25 {\sqrt{15}}^{2} - 25 \sqrt{75} i + 25 i \sqrt{75} - 25 i^{2} {\sqrt{5}}^{2}}$

Step 2.3

Simplify.

$\frac{(\sqrt{5} + i \sqrt{15}) (5 \sqrt{15} - 5 i \sqrt{5})}{500}$

Step 2.4

Cancel the common factor of $5 \sqrt{15} - 5 i \sqrt{5}$ and $500$ .

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Step 2.4.1

Factor $5$ out of $(\sqrt{5} + i \sqrt{15}) (5 \sqrt{15} - 5 i \sqrt{5})$ .

$\frac{5 ((\sqrt{5} + i \sqrt{15}) (\sqrt{15} - i \sqrt{5}))}{500}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $5$ out of $500$ .

$\frac{5 ((\sqrt{5} + i \sqrt{15}) (\sqrt{15} - i \sqrt{5}))}{5 (100)}$

Step 2.4.2.2

Cancel the common factor.

$\frac{5 ((\sqrt{5} + i \sqrt{15}) (\sqrt{15} - i \sqrt{5}))}{5 \cdot 100}$

Step 2.4.2.3

Rewrite the expression.

$\frac{(\sqrt{5} + i \sqrt{15}) (\sqrt{15} - i \sqrt{5})}{100}$

Step 3

Expand $(\sqrt{5} + i \sqrt{15}) (\sqrt{15} - i \sqrt{5})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{\sqrt{5} (\sqrt{15} - i \sqrt{5}) + i \sqrt{15} (\sqrt{15} - i \sqrt{5})}{100}$

Step 3.2

Apply the distributive property.

$\frac{\sqrt{5} \sqrt{15} + \sqrt{5} (- i \sqrt{5}) + i \sqrt{15} (\sqrt{15} - i \sqrt{5})}{100}$

Step 3.3

Apply the distributive property.

$\frac{\sqrt{5} \sqrt{15} + \sqrt{5} (- i \sqrt{5}) + i \sqrt{15} \sqrt{15} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Combine using the product rule for radicals.

$\frac{\sqrt{5 \cdot 15} + \sqrt{5} (- i \sqrt{5}) + i \sqrt{15} \sqrt{15} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.2

Multiply $5$ by $15$ .

$\frac{\sqrt{75} + \sqrt{5} (- i \sqrt{5}) + i \sqrt{15} \sqrt{15} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.3

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 4.1.3.1

Factor $25$ out of $75$ .

$\frac{\sqrt{25 (3)} + \sqrt{5} (- i \sqrt{5}) + i \sqrt{15} \sqrt{15} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.3.2

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{5^{2} \cdot 3} + \sqrt{5} (- i \sqrt{5}) + i \sqrt{15} \sqrt{15} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.4

Pull terms out from under the radical.

$\frac{5 \sqrt{3} + \sqrt{5} (- i \sqrt{5}) + i \sqrt{15} \sqrt{15} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.5

Multiply $\sqrt{5} (- i \sqrt{5})$ .

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Step 4.1.5.1

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{5 \sqrt{3} - i ({\sqrt{5}}^{1} \sqrt{5}) + i \sqrt{15} \sqrt{15} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.5.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{5 \sqrt{3} - i ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) + i \sqrt{15} \sqrt{15} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.5.3

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

$\frac{5 \sqrt{3} - i {\sqrt{5}}^{1 + 1} + i \sqrt{15} \sqrt{15} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.5.4

Add $1$ and $1$ .

$\frac{5 \sqrt{3} - i {\sqrt{5}}^{2} + i \sqrt{15} \sqrt{15} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{5 \sqrt{3} - i {(5^{\frac{1}{2}})}^{2} + i \sqrt{15} \sqrt{15} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{5 \sqrt{3} - i \cdot 5^{\frac{1}{2} \cdot 2} + i \sqrt{15} \sqrt{15} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{5 \sqrt{3} - i \cdot 5^{\frac{2}{2}} + i \sqrt{15} \sqrt{15} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.6.4

Cancel the common factor of $2$ .

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Step 4.1.6.4.1

Cancel the common factor.

$\frac{5 \sqrt{3} - i \cdot 5^{\frac{2}{2}} + i \sqrt{15} \sqrt{15} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.6.4.2

Rewrite the expression.

$\frac{5 \sqrt{3} - i \cdot 5^{1} + i \sqrt{15} \sqrt{15} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.6.5

Evaluate the exponent.

$\frac{5 \sqrt{3} - i \cdot 5 + i \sqrt{15} \sqrt{15} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.7

Multiply $5$ by $- 1$ .

$\frac{5 \sqrt{3} - 5 i + i \sqrt{15} \sqrt{15} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.8

Multiply $i \sqrt{15} \sqrt{15}$ .

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Step 4.1.8.1

Raise $\sqrt{15}$ to the power of $1$ .

$\frac{5 \sqrt{3} - 5 i + i ({\sqrt{15}}^{1} \sqrt{15}) + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.8.2

Raise $\sqrt{15}$ to the power of $1$ .

$\frac{5 \sqrt{3} - 5 i + i ({\sqrt{15}}^{1} {\sqrt{15}}^{1}) + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.8.3

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

$\frac{5 \sqrt{3} - 5 i + i {\sqrt{15}}^{1 + 1} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.8.4

Add $1$ and $1$ .

$\frac{5 \sqrt{3} - 5 i + i {\sqrt{15}}^{2} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.9

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 4.1.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\frac{5 \sqrt{3} - 5 i + i {(15^{\frac{1}{2}})}^{2} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.9.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{5 \sqrt{3} - 5 i + i \cdot 15^{\frac{1}{2} \cdot 2} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.9.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{5 \sqrt{3} - 5 i + i \cdot 15^{\frac{2}{2}} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.9.4

Cancel the common factor of $2$ .

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Step 4.1.9.4.1

Cancel the common factor.

$\frac{5 \sqrt{3} - 5 i + i \cdot 15^{\frac{2}{2}} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.9.4.2

Rewrite the expression.

$\frac{5 \sqrt{3} - 5 i + i \cdot 15^{1} + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.9.5

Evaluate the exponent.

$\frac{5 \sqrt{3} - 5 i + i \cdot 15 + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.10

Move $15$ to the left of $i$ .

$\frac{5 \sqrt{3} - 5 i + 15 \cdot i + i \sqrt{15} (- i \sqrt{5})}{100}$

Step 4.1.11

Multiply $i \sqrt{15} (- i \sqrt{5})$ .

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Step 4.1.11.1

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

$\frac{5 \sqrt{3} - 5 i + 15 i + \sqrt{15} (- (i^{1} i) \sqrt{5})}{100}$

Step 4.1.11.2

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

$\frac{5 \sqrt{3} - 5 i + 15 i + \sqrt{15} (- (i^{1} i^{1}) \sqrt{5})}{100}$

Step 4.1.11.3

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

$\frac{5 \sqrt{3} - 5 i + 15 i + \sqrt{15} (- i^{1 + 1} \sqrt{5})}{100}$

Step 4.1.11.4

Add $1$ and $1$ .

$\frac{5 \sqrt{3} - 5 i + 15 i + \sqrt{15} (- i^{2} \sqrt{5})}{100}$

Step 4.1.11.5

Combine using the product rule for radicals.

$\frac{5 \sqrt{3} - 5 i + 15 i - i^{2} \sqrt{15 \cdot 5}}{100}$

Step 4.1.11.6

Multiply $15$ by $5$ .

$\frac{5 \sqrt{3} - 5 i + 15 i - i^{2} \sqrt{75}}{100}$

Step 4.1.12

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

$\frac{5 \sqrt{3} - 5 i + 15 i - - 1 \sqrt{75}}{100}$

Step 4.1.13

Multiply $- 1$ by $- 1$ .

$\frac{5 \sqrt{3} - 5 i + 15 i + 1 \sqrt{75}}{100}$

Step 4.1.14

Multiply $\sqrt{75}$ by $1$ .

$\frac{5 \sqrt{3} - 5 i + 15 i + \sqrt{75}}{100}$

Step 4.1.15

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 4.1.15.1

Factor $25$ out of $75$ .

$\frac{5 \sqrt{3} - 5 i + 15 i + \sqrt{25 (3)}}{100}$

Step 4.1.15.2

Rewrite $25$ as $5^{2}$ .

$\frac{5 \sqrt{3} - 5 i + 15 i + \sqrt{5^{2} \cdot 3}}{100}$

Step 4.1.16

Pull terms out from under the radical.

$\frac{5 \sqrt{3} - 5 i + 15 i + 5 \sqrt{3}}{100}$

Step 4.2

Add $5 \sqrt{3}$ and $5 \sqrt{3}$ .

$\frac{- 5 i + 15 i + 10 \sqrt{3}}{100}$

Step 4.3

Add $- 5 i$ and $15 i$ .

$\frac{10 i + 10 \sqrt{3}}{100}$

Step 5

Reorder $10 i$ and $10 \sqrt{3}$ .

$\frac{10 \sqrt{3} + 10 i}{100}$

Step 6

Cancel the common factor of $10 \sqrt{3} + 10 i$ and $100$ .

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Step 6.1

Factor $10$ out of $10 \sqrt{3}$ .

$\frac{10 (\sqrt{3}) + 10 i}{100}$

Step 6.2

Factor $10$ out of $10 i$ .

$\frac{10 (\sqrt{3}) + 10 (i)}{100}$

Step 6.3

Factor $10$ out of $10 (\sqrt{3}) + 10 (i)$ .

$\frac{10 (\sqrt{3} + i)}{100}$

Step 6.4

Cancel the common factors.

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Step 6.4.1

Factor $10$ out of $100$ .

$\frac{10 (\sqrt{3} + i)}{10 (10)}$

Step 6.4.2

Cancel the common factor.

$\frac{10 (\sqrt{3} + i)}{10 \cdot 10}$

Step 6.4.3

Rewrite the expression.

$\frac{\sqrt{3} + i}{10}$