Trigonometry Examples

$v ({(\frac{\sqrt{2}}{2})}^{2} + {(\frac{\sqrt{2}}{2})}^{2})$

Step 1

Simplify each term.

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Step 1.1

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

$v (\frac{{\sqrt{2}}^{2}}{2^{2}} + {(\frac{\sqrt{2}}{2})}^{2})$

Step 1.2

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

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Step 1.2.1

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

$v (\frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}} + {(\frac{\sqrt{2}}{2})}^{2})$

Step 1.2.2

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

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

Step 1.2.3

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

$v (\frac{2^{\frac{2}{2}}}{2^{2}} + {(\frac{\sqrt{2}}{2})}^{2})$

Step 1.2.4

Cancel the common factor of $2$ .

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Step 1.2.4.1

Cancel the common factor.

$v (\frac{2^{\frac{2}{2}}}{2^{2}} + {(\frac{\sqrt{2}}{2})}^{2})$

Step 1.2.4.2

Rewrite the expression.

$v (\frac{2^{1}}{2^{2}} + {(\frac{\sqrt{2}}{2})}^{2})$

Step 1.2.5

Evaluate the exponent.

$v (\frac{2}{2^{2}} + {(\frac{\sqrt{2}}{2})}^{2})$

Step 1.3

Raise $2$ to the power of $2$ .

$v (\frac{2}{4} + {(\frac{\sqrt{2}}{2})}^{2})$

Step 1.4

Cancel the common factor of $2$ and $4$ .

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Step 1.4.1

Factor $2$ out of $2$ .

$v (\frac{2 (1)}{4} + {(\frac{\sqrt{2}}{2})}^{2})$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Factor $2$ out of $4$ .

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

Step 1.4.2.2

Cancel the common factor.

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

Step 1.4.2.3

Rewrite the expression.

$v (\frac{1}{2} + {(\frac{\sqrt{2}}{2})}^{2})$

Step 1.5

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

$v (\frac{1}{2} + \frac{{\sqrt{2}}^{2}}{2^{2}})$

Step 1.6

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

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Step 1.6.1

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

$v (\frac{1}{2} + \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}})$

Step 1.6.2

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

$v (\frac{1}{2} + \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}})$

Step 1.6.3

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

$v (\frac{1}{2} + \frac{2^{\frac{2}{2}}}{2^{2}})$

Step 1.6.4

Cancel the common factor of $2$ .

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Step 1.6.4.1

Cancel the common factor.

$v (\frac{1}{2} + \frac{2^{\frac{2}{2}}}{2^{2}})$

Step 1.6.4.2

Rewrite the expression.

$v (\frac{1}{2} + \frac{2^{1}}{2^{2}})$

Step 1.6.5

Evaluate the exponent.

$v (\frac{1}{2} + \frac{2}{2^{2}})$

Step 1.7

Raise $2$ to the power of $2$ .

$v (\frac{1}{2} + \frac{2}{4})$

Step 1.8

Cancel the common factor of $2$ and $4$ .

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Step 1.8.1

Factor $2$ out of $2$ .

$v (\frac{1}{2} + \frac{2 (1)}{4})$

Step 1.8.2

Cancel the common factors.

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Step 1.8.2.1

Factor $2$ out of $4$ .

$v (\frac{1}{2} + \frac{2 \cdot 1}{2 \cdot 2})$

Step 1.8.2.2

Cancel the common factor.

$v (\frac{1}{2} + \frac{2 \cdot 1}{2 \cdot 2})$

Step 1.8.2.3

Rewrite the expression.

$v (\frac{1}{2} + \frac{1}{2})$

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

$v \frac{1 + 1}{2}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Add $1$ and $1$ .

$v \frac{2}{2}$

Step 2.2.2

Divide $2$ by $2$ .

$v \cdot 1$

Step 2.2.3

Multiply $v$ by $1$ .

$v$