Trigonometry Examples

$\frac{(\frac{41}{5 \sqrt{41}}) - (\frac{5}{4})}{1 + (\frac{41}{5 \sqrt{41}}) (\frac{5}{4})}$

Step 1

Multiply the numerator and denominator of the fraction by $20 \sqrt{41}$ .

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Step 1.1

Multiply $\frac{\frac{41}{5 \sqrt{41}} - \frac{5}{4}}{1 + \frac{41}{5 \sqrt{41}} \cdot \frac{5}{4}}$ by $\frac{20 \sqrt{41}}{20 \sqrt{41}}$ .

$\frac{20 \sqrt{41}}{20 \sqrt{41}} \cdot \frac{\frac{41}{5 \sqrt{41}} - \frac{5}{4}}{1 + \frac{41}{5 \sqrt{41}} \cdot \frac{5}{4}}$

Step 1.2

Combine.

$\frac{20 \sqrt{41} (\frac{41}{5 \sqrt{41}} - \frac{5}{4})}{20 \sqrt{41} (1 + \frac{41}{5 \sqrt{41}} \cdot \frac{5}{4})}$

Step 2

Apply the distributive property.

$\frac{20 \sqrt{41} \frac{41}{5 \sqrt{41}} + 20 \sqrt{41} (- \frac{5}{4})}{20 \sqrt{41} \cdot 1 + 20 \sqrt{41} (\frac{41}{5 \sqrt{41}} \cdot \frac{5}{4})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $5 \sqrt{41}$ .

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Step 3.1.1

Factor $5 \sqrt{41}$ out of $20 \sqrt{41}$ .

$\frac{5 \sqrt{41} (4) \frac{41}{5 \sqrt{41}} + 20 \sqrt{41} (- \frac{5}{4})}{20 \sqrt{41} \cdot 1 + 20 \sqrt{41} (\frac{41}{5 \sqrt{41}} \cdot \frac{5}{4})}$

Step 3.1.2

Cancel the common factor.

$\frac{5 \sqrt{41} \cdot 4 \frac{41}{5 \sqrt{41}} + 20 \sqrt{41} (- \frac{5}{4})}{20 \sqrt{41} \cdot 1 + 20 \sqrt{41} (\frac{41}{5 \sqrt{41}} \cdot \frac{5}{4})}$

Step 3.1.3

Rewrite the expression.

$\frac{4 \cdot 41 + 20 \sqrt{41} (- \frac{5}{4})}{20 \sqrt{41} \cdot 1 + 20 \sqrt{41} (\frac{41}{5 \sqrt{41}} \cdot \frac{5}{4})}$

Step 3.2

Multiply $4$ by $41$ .

$\frac{164 + 20 \sqrt{41} (- \frac{5}{4})}{20 \sqrt{41} \cdot 1 + 20 \sqrt{41} (\frac{41}{5 \sqrt{41}} \cdot \frac{5}{4})}$

Step 3.3

Cancel the common factor of $4$ .

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Step 3.3.1

Move the leading negative in $- \frac{5}{4}$ into the numerator.

$\frac{164 + 20 \sqrt{41} \frac{- 5}{4}}{20 \sqrt{41} \cdot 1 + 20 \sqrt{41} (\frac{41}{5 \sqrt{41}} \cdot \frac{5}{4})}$

Step 3.3.2

Factor $4$ out of $20 \sqrt{41}$ .

$\frac{164 + 4 (5 \sqrt{41}) \frac{- 5}{4}}{20 \sqrt{41} \cdot 1 + 20 \sqrt{41} (\frac{41}{5 \sqrt{41}} \cdot \frac{5}{4})}$

Step 3.3.3

Cancel the common factor.

$\frac{164 + 4 (5 \sqrt{41}) \frac{- 5}{4}}{20 \sqrt{41} \cdot 1 + 20 \sqrt{41} (\frac{41}{5 \sqrt{41}} \cdot \frac{5}{4})}$

Step 3.3.4

Rewrite the expression.

$\frac{164 + 5 \sqrt{41} \cdot - 5}{20 \sqrt{41} \cdot 1 + 20 \sqrt{41} (\frac{41}{5 \sqrt{41}} \cdot \frac{5}{4})}$

Step 3.4

Multiply $- 5$ by $5$ .

$\frac{164 - 25 \sqrt{41}}{20 \sqrt{41} \cdot 1 + 20 \sqrt{41} (\frac{41}{5 \sqrt{41}} \cdot \frac{5}{4})}$

Step 4

Simplify the denominator.

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Step 4.1

Multiply $20$ by $1$ .

$\frac{164 - 25 \sqrt{41}}{20 \sqrt{41} + 20 \sqrt{41} \frac{41}{5 \sqrt{41}} \frac{5}{4}}$

Step 4.2

Cancel the common factor of $4$ .

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Step 4.2.1

Factor $4$ out of $20 \sqrt{41} \frac{41}{5 \sqrt{41}}$ .

$\frac{164 - 25 \sqrt{41}}{20 \sqrt{41} + 4 (5 \sqrt{41} \frac{41}{5 \sqrt{41}}) \frac{5}{4}}$

Step 4.2.2

Cancel the common factor.

$\frac{164 - 25 \sqrt{41}}{20 \sqrt{41} + 4 (5 \sqrt{41} \frac{41}{5 \sqrt{41}}) \frac{5}{4}}$

Step 4.2.3

Rewrite the expression.

$\frac{164 - 25 \sqrt{41}}{20 \sqrt{41} + 5 \sqrt{41} \frac{41}{5 \sqrt{41}} \cdot 5}$

Step 4.3

Combine $\frac{41}{5 \sqrt{41}}$ and $5$ .

$\frac{164 - 25 \sqrt{41}}{20 \sqrt{41} + \frac{41 \cdot 5}{5 \sqrt{41}} \sqrt{41} \cdot 5}$

Step 4.4

Multiply $41$ by $5$ .

$\frac{164 - 25 \sqrt{41}}{20 \sqrt{41} + \frac{205}{5 \sqrt{41}} \sqrt{41} \cdot 5}$

Step 4.5

Combine $\frac{205}{5 \sqrt{41}}$ and $\sqrt{41}$ .

$\frac{164 - 25 \sqrt{41}}{20 \sqrt{41} + \frac{205 \sqrt{41}}{5 \sqrt{41}} \cdot 5}$

Step 4.6

Combine $\frac{205 \sqrt{41}}{5 \sqrt{41}}$ and $5$ .

$\frac{164 - 25 \sqrt{41}}{20 \sqrt{41} + \frac{205 \sqrt{41} \cdot 5}{5 \sqrt{41}}}$

Step 4.7

Multiply $5$ by $205$ .

$\frac{164 - 25 \sqrt{41}}{20 \sqrt{41} + \frac{1025 \sqrt{41}}{5 \sqrt{41}}}$

Step 4.8

Cancel the common factor of $1025$ and $5$ .

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Step 4.8.1

Factor $5$ out of $1025 \sqrt{41}$ .

$\frac{164 - 25 \sqrt{41}}{20 \sqrt{41} + \frac{5 (205 \sqrt{41})}{5 \sqrt{41}}}$

Step 4.8.2

Cancel the common factors.

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Step 4.8.2.1

Factor $5$ out of $5 \sqrt{41}$ .

$\frac{164 - 25 \sqrt{41}}{20 \sqrt{41} + \frac{5 (205 \sqrt{41})}{5 (\sqrt{41})}}$

Step 4.8.2.2

Cancel the common factor.

$\frac{164 - 25 \sqrt{41}}{20 \sqrt{41} + \frac{5 (205 \sqrt{41})}{5 \sqrt{41}}}$

Step 4.8.2.3

Rewrite the expression.

$\frac{164 - 25 \sqrt{41}}{20 \sqrt{41} + \frac{205 \sqrt{41}}{\sqrt{41}}}$

Step 4.9

Cancel the common factor of $\sqrt{41}$ .

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Step 4.9.1

Cancel the common factor.

$\frac{164 - 25 \sqrt{41}}{20 \sqrt{41} + \frac{205 \sqrt{41}}{\sqrt{41}}}$

Step 4.9.2

Divide $205$ by $1$ .

$\frac{164 - 25 \sqrt{41}}{20 \sqrt{41} + 205}$

Step 5

Multiply $\frac{164 - 25 \sqrt{41}}{20 \sqrt{41} + 205}$ by $\frac{20 \sqrt{41} - 205}{20 \sqrt{41} - 205}$ .

$\frac{164 - 25 \sqrt{41}}{20 \sqrt{41} + 205} \cdot \frac{20 \sqrt{41} - 205}{20 \sqrt{41} - 205}$

Step 6

Simplify terms.

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Step 6.1

Multiply $\frac{164 - 25 \sqrt{41}}{20 \sqrt{41} + 205}$ by $\frac{20 \sqrt{41} - 205}{20 \sqrt{41} - 205}$ .

$\frac{(164 - 25 \sqrt{41}) (20 \sqrt{41} - 205)}{(20 \sqrt{41} + 205) (20 \sqrt{41} - 205)}$

Step 6.2

Expand the denominator using the FOIL method.

$\frac{(164 - 25 \sqrt{41}) (20 \sqrt{41} - 205)}{400 {\sqrt{41}}^{2} - 4100 \sqrt{41} + 4100 \sqrt{41} - 42025}$

Step 6.3

Simplify.

$\frac{(164 - 25 \sqrt{41}) (20 \sqrt{41} - 205)}{- 25625}$

Step 6.4

Cancel the common factor of $20 \sqrt{41} - 205$ and $- 25625$ .

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Step 6.4.1

Factor $5$ out of $(164 - 25 \sqrt{41}) (20 \sqrt{41} - 205)$ .

$\frac{5 ((164 - 25 \sqrt{41}) (4 \sqrt{41} - 41))}{- 25625}$

Step 6.4.2

Cancel the common factors.

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Step 6.4.2.1

Factor $5$ out of $- 25625$ .

$\frac{5 ((164 - 25 \sqrt{41}) (4 \sqrt{41} - 41))}{5 (- 5125)}$

Step 6.4.2.2

Cancel the common factor.

$\frac{5 ((164 - 25 \sqrt{41}) (4 \sqrt{41} - 41))}{5 \cdot - 5125}$

Step 6.4.2.3

Rewrite the expression.

$\frac{(164 - 25 \sqrt{41}) (4 \sqrt{41} - 41)}{- 5125}$

Step 7

Expand $(164 - 25 \sqrt{41}) (4 \sqrt{41} - 41)$ using the FOIL Method.

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Step 7.1

Apply the distributive property.

$\frac{164 (4 \sqrt{41} - 41) - 25 \sqrt{41} (4 \sqrt{41} - 41)}{- 5125}$

Step 7.2

Apply the distributive property.

$\frac{164 (4 \sqrt{41}) + 164 \cdot - 41 - 25 \sqrt{41} (4 \sqrt{41} - 41)}{- 5125}$

Step 7.3

Apply the distributive property.

$\frac{164 (4 \sqrt{41}) + 164 \cdot - 41 - 25 \sqrt{41} (4 \sqrt{41}) - 25 \sqrt{41} \cdot - 41}{- 5125}$

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $4$ by $164$ .

$\frac{656 \sqrt{41} + 164 \cdot - 41 - 25 \sqrt{41} (4 \sqrt{41}) - 25 \sqrt{41} \cdot - 41}{- 5125}$

Step 8.1.2

Multiply $164$ by $- 41$ .

$\frac{656 \sqrt{41} - 6724 - 25 \sqrt{41} (4 \sqrt{41}) - 25 \sqrt{41} \cdot - 41}{- 5125}$

Step 8.1.3

Multiply $- 25 \sqrt{41} (4 \sqrt{41})$ .

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Step 8.1.3.1

Multiply $4$ by $- 25$ .

$\frac{656 \sqrt{41} - 6724 - 100 \sqrt{41} \sqrt{41} - 25 \sqrt{41} \cdot - 41}{- 5125}$

Step 8.1.3.2

Raise $\sqrt{41}$ to the power of $1$ .

$\frac{656 \sqrt{41} - 6724 - 100 ({\sqrt{41}}^{1} \sqrt{41}) - 25 \sqrt{41} \cdot - 41}{- 5125}$

Step 8.1.3.3

Raise $\sqrt{41}$ to the power of $1$ .

$\frac{656 \sqrt{41} - 6724 - 100 ({\sqrt{41}}^{1} {\sqrt{41}}^{1}) - 25 \sqrt{41} \cdot - 41}{- 5125}$

Step 8.1.3.4

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

$\frac{656 \sqrt{41} - 6724 - 100 {\sqrt{41}}^{1 + 1} - 25 \sqrt{41} \cdot - 41}{- 5125}$

Step 8.1.3.5

Add $1$ and $1$ .

$\frac{656 \sqrt{41} - 6724 - 100 {\sqrt{41}}^{2} - 25 \sqrt{41} \cdot - 41}{- 5125}$

Step 8.1.4

Rewrite ${\sqrt{41}}^{2}$ as $41$ .

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Step 8.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{41}$ as $41^{\frac{1}{2}}$ .

$\frac{656 \sqrt{41} - 6724 - 100 {(41^{\frac{1}{2}})}^{2} - 25 \sqrt{41} \cdot - 41}{- 5125}$

Step 8.1.4.2

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

$\frac{656 \sqrt{41} - 6724 - 100 \cdot 41^{\frac{1}{2} \cdot 2} - 25 \sqrt{41} \cdot - 41}{- 5125}$

Step 8.1.4.3

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

$\frac{656 \sqrt{41} - 6724 - 100 \cdot 41^{\frac{2}{2}} - 25 \sqrt{41} \cdot - 41}{- 5125}$

Step 8.1.4.4

Cancel the common factor of $2$ .

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Step 8.1.4.4.1

Cancel the common factor.

$\frac{656 \sqrt{41} - 6724 - 100 \cdot 41^{\frac{2}{2}} - 25 \sqrt{41} \cdot - 41}{- 5125}$

Step 8.1.4.4.2

Rewrite the expression.

$\frac{656 \sqrt{41} - 6724 - 100 \cdot 41^{1} - 25 \sqrt{41} \cdot - 41}{- 5125}$

Step 8.1.4.5

Evaluate the exponent.

$\frac{656 \sqrt{41} - 6724 - 100 \cdot 41 - 25 \sqrt{41} \cdot - 41}{- 5125}$

Step 8.1.5

Multiply $- 100$ by $41$ .

$\frac{656 \sqrt{41} - 6724 - 4100 - 25 \sqrt{41} \cdot - 41}{- 5125}$

Step 8.1.6

Multiply $- 41$ by $- 25$ .

$\frac{656 \sqrt{41} - 6724 - 4100 + 1025 \sqrt{41}}{- 5125}$

Step 8.2

Add $656 \sqrt{41}$ and $1025 \sqrt{41}$ .

$\frac{- 6724 - 4100 + 1681 \sqrt{41}}{- 5125}$

Step 8.3

Subtract $4100$ from $- 6724$ .

$\frac{- 10824 + 1681 \sqrt{41}}{- 5125}$

Step 9

Cancel the common factor of $- 10824 + 1681 \sqrt{41}$ and $- 5125$ .

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Step 9.1

Factor $41$ out of $- 10824$ .

$\frac{41 (- 264) + 1681 \sqrt{41}}{- 5125}$

Step 9.2

Factor $41$ out of $1681 \sqrt{41}$ .

$\frac{41 (- 264) + 41 (41 \sqrt{41})}{- 5125}$

Step 9.3

Factor $41$ out of $41 (- 264) + 41 (41 \sqrt{41})$ .

$\frac{41 (- 264 + 41 \sqrt{41})}{- 5125}$

Step 9.4

Cancel the common factors.

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Step 9.4.1

Factor $41$ out of $- 5125$ .

$\frac{41 (- 264 + 41 \sqrt{41})}{41 \cdot - 125}$

Step 9.4.2

Cancel the common factor.

$\frac{41 (- 264 + 41 \sqrt{41})}{41 \cdot - 125}$

Step 9.4.3

Rewrite the expression.

$\frac{- 264 + 41 \sqrt{41}}{- 125}$

Step 10

Move the negative in front of the fraction.

$- \frac{- 264 + 41 \sqrt{41}}{125}$

Step 11

Rewrite $- 264$ as $- 1 (264)$ .

$- \frac{- 1 (264) + 41 \sqrt{41}}{125}$

Step 12

Factor $- 1$ out of $41 \sqrt{41}$ .

$- \frac{- 1 (264) - (- 41 \sqrt{41})}{125}$

Step 13

Factor $- 1$ out of $- 1 (264) - (- 41 \sqrt{41})$ .

$- \frac{- 1 (264 - 41 \sqrt{41})}{125}$

Step 14

Simplify the expression.

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Step 14.1

Move the negative in front of the fraction.

$- - \frac{264 - 41 \sqrt{41}}{125}$

Step 14.2

Multiply $- 1$ by $- 1$ .

$1 \frac{264 - 41 \sqrt{41}}{125}$

Step 14.3

Multiply $\frac{264 - 41 \sqrt{41}}{125}$ by $1$ .

$\frac{264 - 41 \sqrt{41}}{125}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$\frac{264 - 41 \sqrt{41}}{125}$

Decimal Form:

$0.01177525 \dots$