Trigonometry Examples

$\frac{2 \tan (30)}{1 - \tan^{2} (30)}$

Step 1

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{2 \frac{\sqrt{3}}{3}}{1 - \tan^{2} (30)}$

Step 2

Simplify the denominator.

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Step 2.1

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

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

Step 2.2

Apply the product rule to $\frac{\sqrt{3}}{3}$ .

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

Step 2.3

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

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Step 2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Cancel the common factor of $2$ .

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Step 2.3.4.1

Cancel the common factor.

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

Step 2.3.4.2

Rewrite the expression.

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

Step 2.3.5

Evaluate the exponent.

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

Step 2.4

Raise $3$ to the power of $2$ .

$\frac{2 \frac{\sqrt{3}}{3}}{1 - \frac{3}{9}}$

Step 2.5

Cancel the common factor of $3$ and $9$ .

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Step 2.5.1

Factor $3$ out of $3$ .

$\frac{2 \frac{\sqrt{3}}{3}}{1 - \frac{3 (1)}{9}}$

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

Factor $3$ out of $9$ .

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

Step 2.5.2.2

Cancel the common factor.

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

Step 2.5.2.3

Rewrite the expression.

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

Step 2.6

Write $1$ as a fraction with a common denominator.

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

Step 2.7

Combine the numerators over the common denominator.

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

Step 2.8

Subtract $1$ from $3$ .

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

Step 3

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Factor $2$ out of $2 \sqrt{3}$ .

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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

Step 6

Cancel the common factor of $3$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\sqrt{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\sqrt{3}$

Decimal Form:

$1.73205080 \dots$