Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Add $1$ and $1$ .

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

Step 1.2.6

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

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Step 1.2.6.1

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

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

Step 1.2.6.2

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

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

Step 1.2.6.3

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

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

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

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

Step 1.2.6.4.2

Rewrite the expression.

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

Step 1.2.6.5

Evaluate the exponent.

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Combine using the product rule for radicals.

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

Step 1.3.3

Multiply $3$ by $2$ .

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

Step 1.3.4

Multiply $2$ by $2$ .

$\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{1}{\sqrt{2}}$

Step 1.4

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

$\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 1.5

Combine and simplify the denominator.

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Step 1.5.1

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

$\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.5.2

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

$\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.5.3

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

$\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.5.4

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

$\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.5.5

Add $1$ and $1$ .

$\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.5.6

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

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Step 1.5.6.1

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

$\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.5.6.2

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

$\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.5.6.3

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

$\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.5.6.4

Cancel the common factor of $2$ .

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Step 1.5.6.4.1

Cancel the common factor.

$\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.5.6.4.2

Rewrite the expression.

$\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2^{1}}$

Step 1.5.6.5

Evaluate the exponent.

$\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$

Step 1.6

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

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Step 1.6.1

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

$\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}$

Step 1.6.2

Multiply $2$ by $2$ .

$\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$

Step 2

Combine the numerators over the common denominator.

$\frac{\sqrt{6} - \sqrt{2}}{4}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{6} - \sqrt{2}}{4}$

Decimal Form:

$0.25881904 \dots$