Trigonometry Examples

\tan^{2} (30)

$\tan^{2} (30)$

Step 1

The exact value of

\tan (30)

$\tan (30)$ is

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

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

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

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

Step 2

Apply the product rule to

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

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

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

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

Step 3

Rewrite

{\sqrt{3}}^{2}

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

3

$3$ .

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Step 3.1

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}}

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

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

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

Step 3.2

Apply the power rule and multiply exponents,

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

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

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

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

Step 3.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 3.4

Cancel the common factor of

2

$2$ .

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Step 3.4.1

Cancel the common factor.

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

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

Step 3.4.2

Rewrite the expression.

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

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

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

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

Step 3.5

Evaluate the exponent.

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

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

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

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

Step 4

Raise

3

$3$ to the power of

2

$2$ .

\frac{3}{9}

$\frac{3}{9}$

Step 5

Cancel the common factor of

3

$3$ and

9

$9$ .

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Step 5.1

Factor

3

$3$ out of

3

$3$ .

\frac{3 (1)}{9}

$\frac{3 (1)}{9}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor

3

$3$ out of

9

$9$ .

\frac{3 \cdot 1}{3 \cdot 3}

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

Step 5.2.2

Cancel the common factor.

\frac{3 \cdot 1}{3 \cdot 3}

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

Step 5.2.3

Rewrite the expression.

\frac{1}{3}

$\frac{1}{3}$

\frac{1}{3}

$\frac{1}{3}$

\frac{1}{3}

$\frac{1}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

\frac{1}{3}

$\frac{1}{3}$

Decimal Form:

0.\overline{3}

$0.\overline{3}$

[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$