Trigonometry Examples

$\sqrt{\frac{1 - (\frac{2}{\sqrt{29}})}{2}}$

Step 1

Multiply $\frac{2}{\sqrt{29}}$ by $\frac{\sqrt{29}}{\sqrt{29}}$ .

$\sqrt{\frac{1 - (\frac{2}{\sqrt{29}} \cdot \frac{\sqrt{29}}{\sqrt{29}})}{2}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{2}{\sqrt{29}}$ by $\frac{\sqrt{29}}{\sqrt{29}}$ .

$\sqrt{\frac{1 - \frac{2 \sqrt{29}}{\sqrt{29} \sqrt{29}}}{2}}$

Step 2.2

Raise $\sqrt{29}$ to the power of $1$ .

$\sqrt{\frac{1 - \frac{2 \sqrt{29}}{{\sqrt{29}}^{1} \sqrt{29}}}{2}}$

Step 2.3

Raise $\sqrt{29}$ to the power of $1$ .

$\sqrt{\frac{1 - \frac{2 \sqrt{29}}{{\sqrt{29}}^{1} {\sqrt{29}}^{1}}}{2}}$

Step 2.4

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

$\sqrt{\frac{1 - \frac{2 \sqrt{29}}{{\sqrt{29}}^{1 + 1}}}{2}}$

Step 2.5

Add $1$ and $1$ .

$\sqrt{\frac{1 - \frac{2 \sqrt{29}}{{\sqrt{29}}^{2}}}{2}}$

Step 2.6

Rewrite ${\sqrt{29}}^{2}$ as $29$ .

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Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{29}$ as $29^{\frac{1}{2}}$ .

$\sqrt{\frac{1 - \frac{2 \sqrt{29}}{{(29^{\frac{1}{2}})}^{2}}}{2}}$

Step 2.6.2

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

$\sqrt{\frac{1 - \frac{2 \sqrt{29}}{29^{\frac{1}{2} \cdot 2}}}{2}}$

Step 2.6.3

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

$\sqrt{\frac{1 - \frac{2 \sqrt{29}}{29^{\frac{2}{2}}}}{2}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\sqrt{\frac{1 - \frac{2 \sqrt{29}}{29^{\frac{2}{2}}}}{2}}$

Step 2.6.4.2

Rewrite the expression.

$\sqrt{\frac{1 - \frac{2 \sqrt{29}}{29^{1}}}{2}}$

Step 2.6.5

Evaluate the exponent.

$\sqrt{\frac{1 - \frac{2 \sqrt{29}}{29}}{2}}$

Step 3

Simplify the expression.

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Step 3.1

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{\frac{29}{29} - \frac{2 \sqrt{29}}{29}}{2}}$

Step 3.2

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{29 - 2 \sqrt{29}}{29}}{2}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{29 - 2 \sqrt{29}}{29} \cdot \frac{1}{2}}$

Step 5

Multiply $\frac{29 - 2 \sqrt{29}}{29} \cdot \frac{1}{2}$ .

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Step 5.1

Multiply $\frac{29 - 2 \sqrt{29}}{29}$ by $\frac{1}{2}$ .

$\sqrt{\frac{29 - 2 \sqrt{29}}{29 \cdot 2}}$

Step 5.2

Multiply $29$ by $2$ .

$\sqrt{\frac{29 - 2 \sqrt{29}}{58}}$

Step 6

Rewrite $\sqrt{\frac{29 - 2 \sqrt{29}}{58}}$ as $\frac{\sqrt{29 - 2 \sqrt{29}}}{\sqrt{58}}$ .

$\frac{\sqrt{29 - 2 \sqrt{29}}}{\sqrt{58}}$

Step 7

Multiply $\frac{\sqrt{29 - 2 \sqrt{29}}}{\sqrt{58}}$ by $\frac{\sqrt{58}}{\sqrt{58}}$ .

$\frac{\sqrt{29 - 2 \sqrt{29}}}{\sqrt{58}} \cdot \frac{\sqrt{58}}{\sqrt{58}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{\sqrt{29 - 2 \sqrt{29}}}{\sqrt{58}}$ by $\frac{\sqrt{58}}{\sqrt{58}}$ .

$\frac{\sqrt{29 - 2 \sqrt{29}} \sqrt{58}}{\sqrt{58} \sqrt{58}}$

Step 8.2

Raise $\sqrt{58}$ to the power of $1$ .

$\frac{\sqrt{29 - 2 \sqrt{29}} \sqrt{58}}{{\sqrt{58}}^{1} \sqrt{58}}$

Step 8.3

Raise $\sqrt{58}$ to the power of $1$ .

$\frac{\sqrt{29 - 2 \sqrt{29}} \sqrt{58}}{{\sqrt{58}}^{1} {\sqrt{58}}^{1}}$

Step 8.4

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

$\frac{\sqrt{29 - 2 \sqrt{29}} \sqrt{58}}{{\sqrt{58}}^{1 + 1}}$

Step 8.5

Add $1$ and $1$ .

$\frac{\sqrt{29 - 2 \sqrt{29}} \sqrt{58}}{{\sqrt{58}}^{2}}$

Step 8.6

Rewrite ${\sqrt{58}}^{2}$ as $58$ .

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Step 8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{58}$ as $58^{\frac{1}{2}}$ .

$\frac{\sqrt{29 - 2 \sqrt{29}} \sqrt{58}}{{(58^{\frac{1}{2}})}^{2}}$

Step 8.6.2

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

$\frac{\sqrt{29 - 2 \sqrt{29}} \sqrt{58}}{58^{\frac{1}{2} \cdot 2}}$

Step 8.6.3

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

$\frac{\sqrt{29 - 2 \sqrt{29}} \sqrt{58}}{58^{\frac{2}{2}}}$

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

$\frac{\sqrt{29 - 2 \sqrt{29}} \sqrt{58}}{58^{\frac{2}{2}}}$

Step 8.6.4.2

Rewrite the expression.

$\frac{\sqrt{29 - 2 \sqrt{29}} \sqrt{58}}{58^{1}}$

Step 8.6.5

Evaluate the exponent.

$\frac{\sqrt{29 - 2 \sqrt{29}} \sqrt{58}}{58}$

Step 9

Combine using the product rule for radicals.

$\frac{\sqrt{(29 - 2 \sqrt{29}) \cdot 58}}{58}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(29 - 2 \sqrt{29}) \cdot 58}}{58}$

Decimal Form:

$0.56062880 \dots$