Trigonometry Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Combine and simplify the denominator.

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Step 2.3.1

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

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

Step 2.3.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.3.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.3.4

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

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

Step 2.3.5

Add $1$ and $1$ .

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

Step 2.3.6

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

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Step 2.3.6.1

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

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

Step 2.3.6.2

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

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

Step 2.3.6.3

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

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

Step 2.3.6.4

Cancel the common factor of $2$ .

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Step 2.3.6.4.1

Cancel the common factor.

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

Step 2.3.6.4.2

Rewrite the expression.

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

Step 2.3.6.5

Evaluate the exponent.

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

Step 2.4

Simplify the numerator.

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Step 2.4.1

Combine using the product rule for radicals.

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

Step 2.4.2

Multiply $3$ by $2$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{2} (1 \frac{2}{\sqrt{6}})$

Step 4

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

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

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

Multiply $\frac{1}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{1}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{1}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 7.2

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Step 7.3

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

Step 7.4

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

$\frac{\sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Step 7.5

Add $1$ and $1$ .

$\frac{\sqrt{6}}{{\sqrt{6}}^{2}}$

Step 7.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

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

Step 7.6.2

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

$\frac{\sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Step 7.6.3

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

$\frac{\sqrt{6}}{6^{\frac{2}{2}}}$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$\frac{\sqrt{6}}{6^{\frac{2}{2}}}$

Step 7.6.4.2

Rewrite the expression.

$\frac{\sqrt{6}}{6^{1}}$

Step 7.6.5

Evaluate the exponent.

$\frac{\sqrt{6}}{6}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{6}}{6}$

Decimal Form:

$0.40824829 \dots$