Trigonometry Examples

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

Step 1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1

Apply the product rule to $- \frac{2}{5}$ .

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

Step 1.2

Apply the product rule to $\frac{2}{5}$ .

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

Step 2

Raise $- 1$ to the power of $2$ .

$- \sqrt{\frac{1}{1 + 1 \frac{2^{2}}{5^{2}}}}$

Step 3

Multiply $\frac{2^{2}}{5^{2}}$ by $1$ .

$- \sqrt{\frac{1}{1 + \frac{2^{2}}{5^{2}}}}$

Step 4

Raise $2$ to the power of $2$ .

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

Step 5

Raise $5$ to the power of $2$ .

$- \sqrt{\frac{1}{1 + \frac{4}{25}}}$

Step 6

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

$- \sqrt{\frac{1}{\frac{25}{25} + \frac{4}{25}}}$

Step 7

Combine the numerators over the common denominator.

$- \sqrt{\frac{1}{\frac{25 + 4}{25}}}$

Step 8

Add $25$ and $4$ .

$- \sqrt{\frac{1}{\frac{29}{25}}}$

Step 9

Multiply the numerator by the reciprocal of the denominator.

$- \sqrt{1 (\frac{25}{29})}$

Step 10

Multiply $\frac{25}{29}$ by $1$ .

$- \sqrt{\frac{25}{29}}$

Step 11

Rewrite $\sqrt{\frac{25}{29}}$ as $\frac{\sqrt{25}}{\sqrt{29}}$ .

$- \frac{\sqrt{25}}{\sqrt{29}}$

Step 12

Simplify the numerator.

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Step 12.1

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

$- \frac{\sqrt{5^{2}}}{\sqrt{29}}$

Step 12.2

Pull terms out from under the radical, assuming positive real numbers.

$- \frac{5}{\sqrt{29}}$

Step 13

Multiply $\frac{5}{\sqrt{29}}$ by $\frac{\sqrt{29}}{\sqrt{29}}$ .

$- (\frac{5}{\sqrt{29}} \cdot \frac{\sqrt{29}}{\sqrt{29}})$

Step 14

Combine and simplify the denominator.

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Step 14.1

Multiply $\frac{5}{\sqrt{29}}$ by $\frac{\sqrt{29}}{\sqrt{29}}$ .

$- \frac{5 \sqrt{29}}{\sqrt{29} \sqrt{29}}$

Step 14.2

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

$- \frac{5 \sqrt{29}}{{\sqrt{29}}^{1} \sqrt{29}}$

Step 14.3

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

$- \frac{5 \sqrt{29}}{{\sqrt{29}}^{1} {\sqrt{29}}^{1}}$

Step 14.4

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

$- \frac{5 \sqrt{29}}{{\sqrt{29}}^{1 + 1}}$

Step 14.5

Add $1$ and $1$ .

$- \frac{5 \sqrt{29}}{{\sqrt{29}}^{2}}$

Step 14.6

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

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Step 14.6.1

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

$- \frac{5 \sqrt{29}}{{(29^{\frac{1}{2}})}^{2}}$

Step 14.6.2

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

$- \frac{5 \sqrt{29}}{29^{\frac{1}{2} \cdot 2}}$

Step 14.6.3

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

$- \frac{5 \sqrt{29}}{29^{\frac{2}{2}}}$

Step 14.6.4

Cancel the common factor of $2$ .

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Step 14.6.4.1

Cancel the common factor.

$- \frac{5 \sqrt{29}}{29^{\frac{2}{2}}}$

Step 14.6.4.2

Rewrite the expression.

$- \frac{5 \sqrt{29}}{29^{1}}$

Step 14.6.5

Evaluate the exponent.

$- \frac{5 \sqrt{29}}{29}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$- \frac{5 \sqrt{29}}{29}$

Decimal Form:

$- 0.92847669 \dots$