Trigonometry Examples

$\cot (\arctan (\frac{x}{\sqrt{3}}))$

Step 1

Set the argument in $\cot (\arctan (\frac{x}{\sqrt{3}}))$ equal to $π n$ to find where the expression is undefined.

$\arctan (\frac{x}{\sqrt{3}}) = π n$ , for any integer $n$

Step 2

Solve for $x$ .

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Step 2.1

Take the inverse arctangent of both sides of the equation to extract $x$ from inside the arctangent.

$\frac{x}{\sqrt{3}} = \tan (π n)$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

$\frac{x}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \tan (π n)$

Step 2.2.1.2

Combine and simplify the denominator.

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Step 2.2.1.2.1

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

$\frac{x \sqrt{3}}{\sqrt{3} \sqrt{3}} = \tan (π n)$

Step 2.2.1.2.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{x \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} = \tan (π n)$

Step 2.2.1.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{x \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} = \tan (π n)$

Step 2.2.1.2.4

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

$\frac{x \sqrt{3}}{{\sqrt{3}}^{1 + 1}} = \tan (π n)$

Step 2.2.1.2.5

Add $1$ and $1$ .

$\frac{x \sqrt{3}}{{\sqrt{3}}^{2}} = \tan (π n)$

Step 2.2.1.2.6

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

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Step 2.2.1.2.6.1

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

$\frac{x \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} = \tan (π n)$

Step 2.2.1.2.6.2

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

$\frac{x \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} = \tan (π n)$

Step 2.2.1.2.6.3

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

$\frac{x \sqrt{3}}{3^{\frac{2}{2}}} = \tan (π n)$

Step 2.2.1.2.6.4

Cancel the common factor of $2$ .

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Step 2.2.1.2.6.4.1

Cancel the common factor.

$\frac{x \sqrt{3}}{3^{\frac{2}{2}}} = \tan (π n)$

Step 2.2.1.2.6.4.2

Rewrite the expression.

$\frac{x \sqrt{3}}{3^{1}} = \tan (π n)$

Step 2.2.1.2.6.5

Evaluate the exponent.

$\frac{x \sqrt{3}}{3} = \tan (π n)$

Step 2.3

Multiply both sides of the equation by $\frac{3}{\sqrt{3}}$ .

$\frac{3}{\sqrt{3}} \cdot \frac{x \sqrt{3}}{3} = \frac{3}{\sqrt{3}} \tan (π n)$

Step 2.4

Simplify both sides of the equation.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Simplify $\frac{3}{\sqrt{3}} \cdot \frac{x \sqrt{3}}{3}$ .

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Step 2.4.1.1.1

Cancel the common factor of $3$ .

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Step 2.4.1.1.1.1

Cancel the common factor.

$\frac{3}{\sqrt{3}} \cdot \frac{x \sqrt{3}}{3} = \frac{3}{\sqrt{3}} \tan (π n)$

Step 2.4.1.1.1.2

Rewrite the expression.

$\frac{1}{\sqrt{3}} (x \sqrt{3}) = \frac{3}{\sqrt{3}} \tan (π n)$

Step 2.4.1.1.2

Cancel the common factor of $\sqrt{3}$ .

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Step 2.4.1.1.2.1

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

$\frac{1}{\sqrt{3}} (\sqrt{3} x) = \frac{3}{\sqrt{3}} \tan (π n)$

Step 2.4.1.1.2.2

Cancel the common factor.

$\frac{1}{\sqrt{3}} (\sqrt{3} x) = \frac{3}{\sqrt{3}} \tan (π n)$

Step 2.4.1.1.2.3

Rewrite the expression.

$x = \frac{3}{\sqrt{3}} \tan (π n)$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\frac{3}{\sqrt{3}} \tan (π n)$ .

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Step 2.4.2.1.1

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

$x = \frac{3}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \tan (π n)$

Step 2.4.2.1.2

Combine and simplify the denominator.

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Step 2.4.2.1.2.1

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

$x = \frac{3 \sqrt{3}}{\sqrt{3} \sqrt{3}} \tan (π n)$

Step 2.4.2.1.2.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \frac{3 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \tan (π n)$

Step 2.4.2.1.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \frac{3 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \tan (π n)$

Step 2.4.2.1.2.4

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

$x = \frac{3 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} \tan (π n)$

Step 2.4.2.1.2.5

Add $1$ and $1$ .

$x = \frac{3 \sqrt{3}}{{\sqrt{3}}^{2}} \tan (π n)$

Step 2.4.2.1.2.6

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

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Step 2.4.2.1.2.6.1

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

$x = \frac{3 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \tan (π n)$

Step 2.4.2.1.2.6.2

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

$x = \frac{3 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \tan (π n)$

Step 2.4.2.1.2.6.3

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

$x = \frac{3 \sqrt{3}}{3^{\frac{2}{2}}} \tan (π n)$

Step 2.4.2.1.2.6.4

Cancel the common factor of $2$ .

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Step 2.4.2.1.2.6.4.1

Cancel the common factor.

$x = \frac{3 \sqrt{3}}{3^{\frac{2}{2}}} \tan (π n)$

Step 2.4.2.1.2.6.4.2

Rewrite the expression.

$x = \frac{3 \sqrt{3}}{3^{1}} \tan (π n)$

Step 2.4.2.1.2.6.5

Evaluate the exponent.

$x = \frac{3 \sqrt{3}}{3} \tan (π n)$

Step 2.4.2.1.3

Cancel the common factor of $3$ .

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Step 2.4.2.1.3.1

Cancel the common factor.

$x = \frac{3 \sqrt{3}}{3} \tan (π n)$

Step 2.4.2.1.3.2

Divide $\sqrt{3}$ by $1$ .

$x = \sqrt{3} \tan (π n)$

Step 3

The domain is all values of $x$ that make the expression defined.

Set-Builder Notation:

${x | x \neq \sqrt{3} \tan (π n)}$ , for any integer $n$

Step 4