Trigonometry Examples

$\sqrt{\frac{1 - (\frac{1}{9})}{1 + \frac{1}{9}}}$

Step 1

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

$\sqrt{\frac{\frac{9}{9} - \frac{1}{9}}{1 + \frac{1}{9}}}$

Step 2

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{9 - 1}{9}}{1 + \frac{1}{9}}}$

Step 3

Subtract $1$ from $9$ .

$\sqrt{\frac{\frac{8}{9}}{1 + \frac{1}{9}}}$

Step 4

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

$\sqrt{\frac{\frac{8}{9}}{\frac{9}{9} + \frac{1}{9}}}$

Step 5

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{8}{9}}{\frac{9 + 1}{9}}}$

Step 6

Add $9$ and $1$ .

$\sqrt{\frac{\frac{8}{9}}{\frac{10}{9}}}$

Step 7

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{8}{9} \cdot \frac{9}{10}}$

Step 8

Cancel the common factor of $2$ .

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Step 8.1

Factor $2$ out of $8$ .

$\sqrt{\frac{2 (4)}{9} \cdot \frac{9}{10}}$

Step 8.2

Factor $2$ out of $10$ .

$\sqrt{\frac{2 \cdot 4}{9} \cdot \frac{9}{2 \cdot 5}}$

Step 8.3

Cancel the common factor.

$\sqrt{\frac{2 \cdot 4}{9} \cdot \frac{9}{2 \cdot 5}}$

Step 8.4

Rewrite the expression.

$\sqrt{\frac{4}{9} \cdot \frac{9}{5}}$

Step 9

Cancel the common factor of $9$ .

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Step 9.1

Cancel the common factor.

$\sqrt{\frac{4}{9} \cdot \frac{9}{5}}$

Step 9.2

Rewrite the expression.

$\sqrt{4 (\frac{1}{5})}$

Step 10

Combine $4$ and $\frac{1}{5}$ .

$\sqrt{\frac{4}{5}}$

Step 11

Rewrite $\sqrt{\frac{4}{5}}$ as $\frac{\sqrt{4}}{\sqrt{5}}$ .

$\frac{\sqrt{4}}{\sqrt{5}}$

Step 12

Simplify the numerator.

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Step 12.1

Rewrite $4$ as $2^{2}$ .

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

Step 12.2

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

$\frac{2}{\sqrt{5}}$

Step 13

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 14

Combine and simplify the denominator.

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Step 14.1

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{2 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 14.2

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

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

Step 14.3

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

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

Step 14.4

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

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

Step 14.5

Add $1$ and $1$ .

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

Step 14.6

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

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Step 14.6.1

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

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

Step 14.6.2

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

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

Step 14.6.3

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

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

Step 14.6.4

Cancel the common factor of $2$ .

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Step 14.6.4.1

Cancel the common factor.

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

Step 14.6.4.2

Rewrite the expression.

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

Step 14.6.5

Evaluate the exponent.

$\frac{2 \sqrt{5}}{5}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{5}}{5}$

Decimal Form:

$0.89442719 \dots$