Trigonometry Examples

$1 - \sqrt{\frac{3}{36}}$

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $3$ and $36$ .

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Step 1.1.1

Factor $3$ out of $3$ .

$1 - \sqrt{\frac{3 (1)}{36}}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $3$ out of $36$ .

$1 - \sqrt{\frac{3 \cdot 1}{3 \cdot 12}}$

Step 1.1.2.2

Cancel the common factor.

$1 - \sqrt{\frac{3 \cdot 1}{3 \cdot 12}}$

Step 1.1.2.3

Rewrite the expression.

$1 - \sqrt{\frac{1}{12}}$

Step 1.2

Rewrite $\sqrt{\frac{1}{12}}$ as $\frac{\sqrt{1}}{\sqrt{12}}$ .

$1 - \frac{\sqrt{1}}{\sqrt{12}}$

Step 1.3

Any root of $1$ is $1$ .

$1 - \frac{1}{\sqrt{12}}$

Step 1.4

Simplify the denominator.

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Step 1.4.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.4.1.1

Factor $4$ out of $12$ .

$1 - \frac{1}{\sqrt{4 (3)}}$

Step 1.4.1.2

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

$1 - \frac{1}{\sqrt{2^{2} \cdot 3}}$

Step 1.4.2

Pull terms out from under the radical.

$1 - \frac{1}{2 \sqrt{3}}$

Step 1.5

Multiply $\frac{1}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$1 - (\frac{1}{2 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 1.6

Combine and simplify the denominator.

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Step 1.6.1

Multiply $\frac{1}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$1 - \frac{\sqrt{3}}{2 \sqrt{3} \sqrt{3}}$

Step 1.6.2

Move $\sqrt{3}$ .

$1 - \frac{\sqrt{3}}{2 (\sqrt{3} \sqrt{3})}$

Step 1.6.3

Raise $\sqrt{3}$ to the power of $1$ .

$1 - \frac{\sqrt{3}}{2 ({\sqrt{3}}^{1} \sqrt{3})}$

Step 1.6.4

Raise $\sqrt{3}$ to the power of $1$ .

$1 - \frac{\sqrt{3}}{2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

Step 1.6.5

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

$1 - \frac{\sqrt{3}}{2 {\sqrt{3}}^{1 + 1}}$

Step 1.6.6

Add $1$ and $1$ .

$1 - \frac{\sqrt{3}}{2 {\sqrt{3}}^{2}}$

Step 1.6.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.6.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$1 - \frac{\sqrt{3}}{2 {(3^{\frac{1}{2}})}^{2}}$

Step 1.6.7.2

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

$1 - \frac{\sqrt{3}}{2 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 1.6.7.3

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

$1 - \frac{\sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

Step 1.6.7.4

Cancel the common factor of $2$ .

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Step 1.6.7.4.1

Cancel the common factor.

$1 - \frac{\sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

Step 1.6.7.4.2

Rewrite the expression.

$1 - \frac{\sqrt{3}}{2 \cdot 3^{1}}$

Step 1.6.7.5

Evaluate the exponent.

$1 - \frac{\sqrt{3}}{2 \cdot 3}$

Step 1.7

Multiply $2$ by $3$ .

$1 - \frac{\sqrt{3}}{6}$

Step 2

Simplify the expression.

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Step 2.1

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

$\frac{6}{6} - \frac{\sqrt{3}}{6}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{6 - \sqrt{3}}{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{6 - \sqrt{3}}{6}$

Decimal Form:

$0.71132486 \dots$