Trigonometry Examples

$\sqrt{\frac{1 + \frac{- \sqrt{35}}{6}}{2}}$

Step 1

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

$\sqrt{\frac{\frac{6}{6} + \frac{- \sqrt{35}}{6}}{2}}$

Step 2

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{6 - \sqrt{35}}{6}}{2}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{6 - \sqrt{35}}{6} \cdot \frac{1}{2}}$

Step 4

Multiply $\frac{6 - \sqrt{35}}{6} \cdot \frac{1}{2}$ .

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Step 4.1

Multiply $\frac{6 - \sqrt{35}}{6}$ by $\frac{1}{2}$ .

$\sqrt{\frac{6 - \sqrt{35}}{6 \cdot 2}}$

Step 4.2

Multiply $6$ by $2$ .

$\sqrt{\frac{6 - \sqrt{35}}{12}}$

Step 5

Rewrite $\sqrt{\frac{6 - \sqrt{35}}{12}}$ as $\frac{\sqrt{6 - \sqrt{35}}}{\sqrt{12}}$ .

$\frac{\sqrt{6 - \sqrt{35}}}{\sqrt{12}}$

Step 6

Simplify the denominator.

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Step 6.1

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

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Step 6.1.1

Factor $4$ out of $12$ .

$\frac{\sqrt{6 - \sqrt{35}}}{\sqrt{4 (3)}}$

Step 6.1.2

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

$\frac{\sqrt{6 - \sqrt{35}}}{\sqrt{2^{2} \cdot 3}}$

Step 6.2

Pull terms out from under the radical.

$\frac{\sqrt{6 - \sqrt{35}}}{2 \sqrt{3}}$

Step 7

Multiply $\frac{\sqrt{6 - \sqrt{35}}}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{6 - \sqrt{35}}}{2 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{\sqrt{6 - \sqrt{35}}}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{6 - \sqrt{35}} \sqrt{3}}{2 \sqrt{3} \sqrt{3}}$

Step 8.2

Move $\sqrt{3}$ .

$\frac{\sqrt{6 - \sqrt{35}} \sqrt{3}}{2 (\sqrt{3} \sqrt{3})}$

Step 8.3

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

$\frac{\sqrt{6 - \sqrt{35}} \sqrt{3}}{2 ({\sqrt{3}}^{1} \sqrt{3})}$

Step 8.4

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

$\frac{\sqrt{6 - \sqrt{35}} \sqrt{3}}{2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

Step 8.5

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

$\frac{\sqrt{6 - \sqrt{35}} \sqrt{3}}{2 {\sqrt{3}}^{1 + 1}}$

Step 8.6

Add $1$ and $1$ .

$\frac{\sqrt{6 - \sqrt{35}} \sqrt{3}}{2 {\sqrt{3}}^{2}}$

Step 8.7

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

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Step 8.7.1

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

$\frac{\sqrt{6 - \sqrt{35}} \sqrt{3}}{2 {(3^{\frac{1}{2}})}^{2}}$

Step 8.7.2

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

$\frac{\sqrt{6 - \sqrt{35}} \sqrt{3}}{2 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 8.7.3

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

$\frac{\sqrt{6 - \sqrt{35}} \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

Step 8.7.4

Cancel the common factor of $2$ .

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Step 8.7.4.1

Cancel the common factor.

$\frac{\sqrt{6 - \sqrt{35}} \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

Step 8.7.4.2

Rewrite the expression.

$\frac{\sqrt{6 - \sqrt{35}} \sqrt{3}}{2 \cdot 3^{1}}$

Step 8.7.5

Evaluate the exponent.

$\frac{\sqrt{6 - \sqrt{35}} \sqrt{3}}{2 \cdot 3}$

Step 9

Combine using the product rule for radicals.

$\frac{\sqrt{(6 - \sqrt{35}) \cdot 3}}{2 \cdot 3}$

Step 10

Multiply $2$ by $3$ .

$\frac{\sqrt{(6 - \sqrt{35}) \cdot 3}}{6}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(6 - \sqrt{35}) \cdot 3}}{6}$

Decimal Form:

$0.08362626 \dots$