Trigonometry Examples

$\frac{- \frac{1}{\sqrt{15}} + \frac{12}{5}}{1 - \frac{1}{\sqrt{15}} \cdot \frac{12}{5}}$

Step 1

Multiply the numerator and denominator of the fraction by $\sqrt{15} \cdot 5$ .

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Step 1.1

Multiply $\frac{- \frac{1}{\sqrt{15}} + \frac{12}{5}}{1 - \frac{1}{\sqrt{15}} \cdot \frac{12}{5}}$ by $\frac{\sqrt{15} \cdot 5}{\sqrt{15} \cdot 5}$ .

$\frac{\sqrt{15} \cdot 5}{\sqrt{15} \cdot 5} \cdot \frac{- \frac{1}{\sqrt{15}} + \frac{12}{5}}{1 - \frac{1}{\sqrt{15}} \cdot \frac{12}{5}}$

Step 1.2

Combine.

$\frac{\sqrt{15} \cdot 5 (- \frac{1}{\sqrt{15}} + \frac{12}{5})}{\sqrt{15} \cdot 5 (1 - \frac{1}{\sqrt{15}} \cdot \frac{12}{5})}$

Step 2

Apply the distributive property.

$\frac{\sqrt{15} \cdot 5 (- \frac{1}{\sqrt{15}}) + \sqrt{15} \cdot 5 \frac{12}{5}}{\sqrt{15} \cdot 5 \cdot 1 + \sqrt{15} \cdot 5 (- \frac{1}{\sqrt{15}} \cdot \frac{12}{5})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $\sqrt{15}$ .

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Step 3.1.1

Move the leading negative in $- \frac{1}{\sqrt{15}}$ into the numerator.

$\frac{\sqrt{15} \cdot 5 \frac{- 1}{\sqrt{15}} + \sqrt{15} \cdot 5 \frac{12}{5}}{\sqrt{15} \cdot 5 \cdot 1 + \sqrt{15} \cdot 5 (- \frac{1}{\sqrt{15}} \cdot \frac{12}{5})}$

Step 3.1.2

Factor $\sqrt{15}$ out of $\sqrt{15} \cdot 5$ .

$\frac{\sqrt{15} (5) \frac{- 1}{\sqrt{15}} + \sqrt{15} \cdot 5 \frac{12}{5}}{\sqrt{15} \cdot 5 \cdot 1 + \sqrt{15} \cdot 5 (- \frac{1}{\sqrt{15}} \cdot \frac{12}{5})}$

Step 3.1.3

Cancel the common factor.

$\frac{\sqrt{15} \cdot 5 \frac{- 1}{\sqrt{15}} + \sqrt{15} \cdot 5 \frac{12}{5}}{\sqrt{15} \cdot 5 \cdot 1 + \sqrt{15} \cdot 5 (- \frac{1}{\sqrt{15}} \cdot \frac{12}{5})}$

Step 3.1.4

Rewrite the expression.

$\frac{5 \cdot - 1 + \sqrt{15} \cdot 5 \frac{12}{5}}{\sqrt{15} \cdot 5 \cdot 1 + \sqrt{15} \cdot 5 (- \frac{1}{\sqrt{15}} \cdot \frac{12}{5})}$

Step 3.2

Multiply $5$ by $- 1$ .

$\frac{- 5 + \sqrt{15} \cdot 5 \frac{12}{5}}{\sqrt{15} \cdot 5 \cdot 1 + \sqrt{15} \cdot 5 (- \frac{1}{\sqrt{15}} \cdot \frac{12}{5})}$

Step 3.3

Cancel the common factor of $5$ .

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Step 3.3.1

Factor $5$ out of $\sqrt{15} \cdot 5$ .

$\frac{- 5 + 5 \cdot \sqrt{15} \frac{12}{5}}{\sqrt{15} \cdot 5 \cdot 1 + \sqrt{15} \cdot 5 (- \frac{1}{\sqrt{15}} \cdot \frac{12}{5})}$

Step 3.3.2

Cancel the common factor.

$\frac{- 5 + 5 \cdot \sqrt{15} \frac{12}{5}}{\sqrt{15} \cdot 5 \cdot 1 + \sqrt{15} \cdot 5 (- \frac{1}{\sqrt{15}} \cdot \frac{12}{5})}$

Step 3.3.3

Rewrite the expression.

$\frac{- 5 + \sqrt{15} \cdot 12}{\sqrt{15} \cdot 5 \cdot 1 + \sqrt{15} \cdot 5 (- \frac{1}{\sqrt{15}} \cdot \frac{12}{5})}$

Step 4

Move $12$ to the left of $\sqrt{15}$ .

$\frac{- 5 + 12 \sqrt{15}}{\sqrt{15} \cdot 5 \cdot 1 + \sqrt{15} \cdot 5 (- \frac{1}{\sqrt{15}} \cdot \frac{12}{5})}$

Step 5

Simplify the denominator.

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Step 5.1

Move $5$ to the left of $\sqrt{15}$ .

$\frac{- 5 + 12 \sqrt{15}}{5 \cdot \sqrt{15} \cdot 1 + \sqrt{15} \cdot 5 (- \frac{1}{\sqrt{15}} \cdot \frac{12}{5})}$

Step 5.2

Multiply $5$ by $1$ .

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + \sqrt{15} \cdot 5 (- \frac{1}{\sqrt{15}} \cdot \frac{12}{5})}$

Step 5.3

Move $5$ to the left of $\sqrt{15}$ .

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \cdot \sqrt{15} (- \frac{1}{\sqrt{15}} \cdot \frac{12}{5})}$

Step 5.4

Multiply $\frac{1}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} (- (\frac{1}{\sqrt{15}} \cdot \frac{\sqrt{15}}{\sqrt{15}}) \cdot \frac{12}{5})}$

Step 5.5

Combine and simplify the denominator.

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Step 5.5.1

Multiply $\frac{1}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} (- \frac{\sqrt{15}}{\sqrt{15} \sqrt{15}} \cdot \frac{12}{5})}$

Step 5.5.2

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

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} (- \frac{\sqrt{15}}{{\sqrt{15}}^{1} \sqrt{15}} \cdot \frac{12}{5})}$

Step 5.5.3

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

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} (- \frac{\sqrt{15}}{{\sqrt{15}}^{1} {\sqrt{15}}^{1}} \cdot \frac{12}{5})}$

Step 5.5.4

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

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} (- \frac{\sqrt{15}}{{\sqrt{15}}^{1 + 1}} \cdot \frac{12}{5})}$

Step 5.5.5

Add $1$ and $1$ .

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} (- \frac{\sqrt{15}}{{\sqrt{15}}^{2}} \cdot \frac{12}{5})}$

Step 5.5.6

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

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Step 5.5.6.1

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

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} (- \frac{\sqrt{15}}{{(15^{\frac{1}{2}})}^{2}} \cdot \frac{12}{5})}$

Step 5.5.6.2

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

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} (- \frac{\sqrt{15}}{15^{\frac{1}{2} \cdot 2}} \cdot \frac{12}{5})}$

Step 5.5.6.3

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

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} (- \frac{\sqrt{15}}{15^{\frac{2}{2}}} \cdot \frac{12}{5})}$

Step 5.5.6.4

Cancel the common factor of $2$ .

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Step 5.5.6.4.1

Cancel the common factor.

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} (- \frac{\sqrt{15}}{15^{\frac{2}{2}}} \cdot \frac{12}{5})}$

Step 5.5.6.4.2

Rewrite the expression.

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} (- \frac{\sqrt{15}}{15^{1}} \cdot \frac{12}{5})}$

Step 5.5.6.5

Evaluate the exponent.

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} (- \frac{\sqrt{15}}{15} \cdot \frac{12}{5})}$

Step 5.6

Cancel the common factor of $3$ .

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Step 5.6.1

Move the leading negative in $- \frac{\sqrt{15}}{15}$ into the numerator.

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} (\frac{- \sqrt{15}}{15} \cdot \frac{12}{5})}$

Step 5.6.2

Factor $3$ out of $15$ .

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} (\frac{- \sqrt{15}}{3 (5)} \cdot \frac{12}{5})}$

Step 5.6.3

Factor $3$ out of $12$ .

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} (\frac{- \sqrt{15}}{3 \cdot 5} \cdot \frac{3 \cdot 4}{5})}$

Step 5.6.4

Cancel the common factor.

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} (\frac{- \sqrt{15}}{3 \cdot 5} \cdot \frac{3 \cdot 4}{5})}$

Step 5.6.5

Rewrite the expression.

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} (\frac{- \sqrt{15}}{5} \cdot \frac{4}{5})}$

Step 5.7

Multiply $\frac{- \sqrt{15}}{5}$ by $\frac{4}{5}$ .

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} \frac{- \sqrt{15} \cdot 4}{5 \cdot 5}}$

Step 5.8

Multiply $4$ by $- 1$ .

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} \frac{- 4 \sqrt{15}}{5 \cdot 5}}$

Step 5.9

Multiply $5$ by $5$ .

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} \frac{- 4 \sqrt{15}}{25}}$

Step 5.10

Cancel the common factor of $5$ .

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Step 5.10.1

Factor $5$ out of $5 \sqrt{15}$ .

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 (\sqrt{15}) \frac{- 4 \sqrt{15}}{25}}$

Step 5.10.2

Factor $5$ out of $25$ .

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 (\sqrt{15}) \frac{- 4 \sqrt{15}}{5 (5)}}$

Step 5.10.3

Cancel the common factor.

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + 5 \sqrt{15} \frac{- 4 \sqrt{15}}{5 \cdot 5}}$

Step 5.10.4

Rewrite the expression.

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + \sqrt{15} \frac{- 4 \sqrt{15}}{5}}$

Step 5.11

Combine $\sqrt{15}$ and $\frac{- 4 \sqrt{15}}{5}$ .

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + \frac{\sqrt{15} (- 4 \sqrt{15})}{5}}$

Step 5.12

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

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + \frac{- 4 ({\sqrt{15}}^{1} \sqrt{15})}{5}}$

Step 5.13

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

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + \frac{- 4 ({\sqrt{15}}^{1} {\sqrt{15}}^{1})}{5}}$

Step 5.14

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

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + \frac{- 4 {\sqrt{15}}^{1 + 1}}{5}}$

Step 5.15

Add $1$ and $1$ .

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + \frac{- 4 {\sqrt{15}}^{2}}{5}}$

Step 5.16

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

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Step 5.16.1

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

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + \frac{- 4 {(15^{\frac{1}{2}})}^{2}}{5}}$

Step 5.16.2

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

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + \frac{- 4 \cdot 15^{\frac{1}{2} \cdot 2}}{5}}$

Step 5.16.3

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

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + \frac{- 4 \cdot 15^{\frac{2}{2}}}{5}}$

Step 5.16.4

Cancel the common factor of $2$ .

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Step 5.16.4.1

Cancel the common factor.

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + \frac{- 4 \cdot 15^{\frac{2}{2}}}{5}}$

Step 5.16.4.2

Rewrite the expression.

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + \frac{- 4 \cdot 15^{1}}{5}}$

Step 5.16.5

Evaluate the exponent.

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + \frac{- 4 \cdot 15}{5}}$

Step 5.17

Multiply $- 4$ by $15$ .

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} + \frac{- 60}{5}}$

Step 5.18

Divide $- 60$ by $5$ .

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} - 12}$

Step 6

Multiply $\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} - 12}$ by $\frac{5 \sqrt{15} + 12}{5 \sqrt{15} + 12}$ .

$\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} - 12} \cdot \frac{5 \sqrt{15} + 12}{5 \sqrt{15} + 12}$

Step 7

Combine fractions.

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Step 7.1

Multiply $\frac{- 5 + 12 \sqrt{15}}{5 \sqrt{15} - 12}$ by $\frac{5 \sqrt{15} + 12}{5 \sqrt{15} + 12}$ .

$\frac{(- 5 + 12 \sqrt{15}) (5 \sqrt{15} + 12)}{(5 \sqrt{15} - 12) (5 \sqrt{15} + 12)}$

Step 7.2

Expand the denominator using the FOIL method.

$\frac{(- 5 + 12 \sqrt{15}) (5 \sqrt{15} + 12)}{25 {\sqrt{15}}^{2} + 60 \sqrt{15} - 60 \sqrt{15} - 144}$

Step 7.3

Simplify.

$\frac{(- 5 + 12 \sqrt{15}) (5 \sqrt{15} + 12)}{231}$

Step 8

Expand $(- 5 + 12 \sqrt{15}) (5 \sqrt{15} + 12)$ using the FOIL Method.

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Step 8.1

Apply the distributive property.

$\frac{- 5 (5 \sqrt{15} + 12) + 12 \sqrt{15} (5 \sqrt{15} + 12)}{231}$

Step 8.2

Apply the distributive property.

$\frac{- 5 (5 \sqrt{15}) - 5 \cdot 12 + 12 \sqrt{15} (5 \sqrt{15} + 12)}{231}$

Step 8.3

Apply the distributive property.

$\frac{- 5 (5 \sqrt{15}) - 5 \cdot 12 + 12 \sqrt{15} (5 \sqrt{15}) + 12 \sqrt{15} \cdot 12}{231}$

Step 9

Simplify and combine like terms.

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Step 9.1

Simplify each term.

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Step 9.1.1

Multiply $5$ by $- 5$ .

$\frac{- 25 \sqrt{15} - 5 \cdot 12 + 12 \sqrt{15} (5 \sqrt{15}) + 12 \sqrt{15} \cdot 12}{231}$

Step 9.1.2

Multiply $- 5$ by $12$ .

$\frac{- 25 \sqrt{15} - 60 + 12 \sqrt{15} (5 \sqrt{15}) + 12 \sqrt{15} \cdot 12}{231}$

Step 9.1.3

Multiply $12 \sqrt{15} (5 \sqrt{15})$ .

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Step 9.1.3.1

Multiply $5$ by $12$ .

$\frac{- 25 \sqrt{15} - 60 + 60 \sqrt{15} \sqrt{15} + 12 \sqrt{15} \cdot 12}{231}$

Step 9.1.3.2

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

$\frac{- 25 \sqrt{15} - 60 + 60 ({\sqrt{15}}^{1} \sqrt{15}) + 12 \sqrt{15} \cdot 12}{231}$

Step 9.1.3.3

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

$\frac{- 25 \sqrt{15} - 60 + 60 ({\sqrt{15}}^{1} {\sqrt{15}}^{1}) + 12 \sqrt{15} \cdot 12}{231}$

Step 9.1.3.4

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

$\frac{- 25 \sqrt{15} - 60 + 60 {\sqrt{15}}^{1 + 1} + 12 \sqrt{15} \cdot 12}{231}$

Step 9.1.3.5

Add $1$ and $1$ .

$\frac{- 25 \sqrt{15} - 60 + 60 {\sqrt{15}}^{2} + 12 \sqrt{15} \cdot 12}{231}$

Step 9.1.4

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

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Step 9.1.4.1

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

$\frac{- 25 \sqrt{15} - 60 + 60 {(15^{\frac{1}{2}})}^{2} + 12 \sqrt{15} \cdot 12}{231}$

Step 9.1.4.2

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

$\frac{- 25 \sqrt{15} - 60 + 60 \cdot 15^{\frac{1}{2} \cdot 2} + 12 \sqrt{15} \cdot 12}{231}$

Step 9.1.4.3

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

$\frac{- 25 \sqrt{15} - 60 + 60 \cdot 15^{\frac{2}{2}} + 12 \sqrt{15} \cdot 12}{231}$

Step 9.1.4.4

Cancel the common factor of $2$ .

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Step 9.1.4.4.1

Cancel the common factor.

$\frac{- 25 \sqrt{15} - 60 + 60 \cdot 15^{\frac{2}{2}} + 12 \sqrt{15} \cdot 12}{231}$

Step 9.1.4.4.2

Rewrite the expression.

$\frac{- 25 \sqrt{15} - 60 + 60 \cdot 15^{1} + 12 \sqrt{15} \cdot 12}{231}$

Step 9.1.4.5

Evaluate the exponent.

$\frac{- 25 \sqrt{15} - 60 + 60 \cdot 15 + 12 \sqrt{15} \cdot 12}{231}$

Step 9.1.5

Multiply $60$ by $15$ .

$\frac{- 25 \sqrt{15} - 60 + 900 + 12 \sqrt{15} \cdot 12}{231}$

Step 9.1.6

Multiply $12$ by $12$ .

$\frac{- 25 \sqrt{15} - 60 + 900 + 144 \sqrt{15}}{231}$

Step 9.2

Add $- 25 \sqrt{15}$ and $144 \sqrt{15}$ .

$\frac{- 60 + 900 + 119 \sqrt{15}}{231}$

Step 9.3

Add $- 60$ and $900$ .

$\frac{840 + 119 \sqrt{15}}{231}$

Step 10

Cancel the common factor of $840 + 119 \sqrt{15}$ and $231$ .

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Step 10.1

Factor $7$ out of $840$ .

$\frac{7 (120) + 119 \sqrt{15}}{231}$

Step 10.2

Factor $7$ out of $119 \sqrt{15}$ .

$\frac{7 (120) + 7 (17 \sqrt{15})}{231}$

Step 10.3

Factor $7$ out of $7 (120) + 7 (17 \sqrt{15})$ .

$\frac{7 (120 + 17 \sqrt{15})}{231}$

Step 10.4

Cancel the common factors.

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Step 10.4.1

Factor $7$ out of $231$ .

$\frac{7 (120 + 17 \sqrt{15})}{7 \cdot 33}$

Step 10.4.2

Cancel the common factor.

$\frac{7 (120 + 17 \sqrt{15})}{7 \cdot 33}$

Step 10.4.3

Rewrite the expression.

$\frac{120 + 17 \sqrt{15}}{33}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{120 + 17 \sqrt{15}}{33}$

Decimal Form:

$5.63153687 \dots$