Trigonometry Examples

\sqrt{{(\frac{\sqrt{3}}{2})}^{2} + {(\frac{1}{2})}^{2}}

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

Step 1

Apply the product rule to

\frac{\sqrt{3}}{2}

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

\sqrt{\frac{{\sqrt{3}}^{2}}{2^{2}} + {(\frac{1}{2})}^{2}}

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

Step 2

Rewrite

{\sqrt{3}}^{2}

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

3

$3$ .

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Step 2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{3}

$\sqrt{3}$ as

3^{\frac{1}{2}}

$3^{\frac{1}{2}}$ .

\sqrt{\frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}} + {(\frac{1}{2})}^{2}}

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

Step 2.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 2.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\sqrt{\frac{3^{\frac{2}{2}}}{2^{2}} + {(\frac{1}{2})}^{2}}

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

Step 2.4

Cancel the common factor of

2

$2$ .

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Rewrite the expression.

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

Step 2.5

Evaluate the exponent.

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

Step 3

Simplify the expression.

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Step 3.1

Raise

$2$ to the power of

$2$ .

$\sqrt{\frac{3}{4} + {(\frac{1}{2})}^{2}}$

Step 3.2

Apply the product rule to

$\frac{1}{2}$ .

$\sqrt{\frac{3}{4} + \frac{1^{2}}{2^{2}}}$

Step 3.3

One to any power is one.

$\sqrt{\frac{3}{4} + \frac{1}{2^{2}}}$

Step 3.4

Raise

$2$ to the power of

$2$ .

$\sqrt{\frac{3}{4} + \frac{1}{4}}$

Step 3.5

Combine the numerators over the common denominator.

$\sqrt{\frac{3 + 1}{4}}$

Step 3.6

Add

$3$ and

$1$ .

$\sqrt{\frac{4}{4}}$

Step 3.7

Divide

$4$ by

$4$ .

$\sqrt{1}$

Step 3.8

Any root of

$1$ is

$1$ .

$1$