Trigonometry Examples

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\cot (\arctan (\frac{y}{\sqrt{3}})) = x$ .

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

Step 2.2

Take the inverse cotangent of both sides of the equation to extract $\arctan (\frac{y}{\sqrt{3}})$ from inside the cotangent.

$\arctan (\frac{y}{\sqrt{3}}) = arccot (x)$

Step 2.3

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

$\frac{y}{\sqrt{3}} = \tan (arccot (x))$

Step 2.4

Simplify the left side.

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Step 2.4.1

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

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Step 2.4.1.1

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

$\frac{y}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \tan (arccot (x))$

Step 2.4.1.2

Combine and simplify the denominator.

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Step 2.4.1.2.1

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

$\frac{y \sqrt{3}}{\sqrt{3} \sqrt{3}} = \tan (arccot (x))$

Step 2.4.1.2.2

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

$\frac{y \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} = \tan (arccot (x))$

Step 2.4.1.2.3

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

$\frac{y \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} = \tan (arccot (x))$

Step 2.4.1.2.4

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

$\frac{y \sqrt{3}}{{\sqrt{3}}^{1 + 1}} = \tan (arccot (x))$

Step 2.4.1.2.5

Add $1$ and $1$ .

$\frac{y \sqrt{3}}{{\sqrt{3}}^{2}} = \tan (arccot (x))$

Step 2.4.1.2.6

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

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Step 2.4.1.2.6.1

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

$\frac{y \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} = \tan (arccot (x))$

Step 2.4.1.2.6.2

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

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

Step 2.4.1.2.6.3

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

$\frac{y \sqrt{3}}{3^{\frac{2}{2}}} = \tan (arccot (x))$

Step 2.4.1.2.6.4

Cancel the common factor of $2$ .

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Step 2.4.1.2.6.4.1

Cancel the common factor.

$\frac{y \sqrt{3}}{3^{\frac{2}{2}}} = \tan (arccot (x))$

Step 2.4.1.2.6.4.2

Rewrite the expression.

$\frac{y \sqrt{3}}{3^{1}} = \tan (arccot (x))$

Step 2.4.1.2.6.5

Evaluate the exponent.

$\frac{y \sqrt{3}}{3} = \tan (arccot (x))$

Step 2.5

Simplify the right side.

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Step 2.5.1

Draw a triangle in the plane with vertices $(x, 1)$ , $(x, 0)$ , and the origin. Then $arccot (x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(x, 1)$ . Therefore, $\tan (arccot (x))$ is $\frac{1}{x}$ .

$\frac{y \sqrt{3}}{3} = \frac{1}{x}$

Step 2.6

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

$\frac{3}{\sqrt{3}} \cdot \frac{y \sqrt{3}}{3} = \frac{3}{\sqrt{3}} \cdot \frac{1}{x}$

Step 2.7

Simplify both sides of the equation.

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Step 2.7.1

Simplify the left side.

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Step 2.7.1.1

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

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Step 2.7.1.1.1

Cancel the common factor of $3$ .

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Step 2.7.1.1.1.1

Cancel the common factor.

$\frac{3}{\sqrt{3}} \cdot \frac{y \sqrt{3}}{3} = \frac{3}{\sqrt{3}} \cdot \frac{1}{x}$

Step 2.7.1.1.1.2

Rewrite the expression.

$\frac{1}{\sqrt{3}} (y \sqrt{3}) = \frac{3}{\sqrt{3}} \cdot \frac{1}{x}$

Step 2.7.1.1.2

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

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Step 2.7.1.1.2.1

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

$\frac{1}{\sqrt{3}} (\sqrt{3} y) = \frac{3}{\sqrt{3}} \cdot \frac{1}{x}$

Step 2.7.1.1.2.2

Cancel the common factor.

$\frac{1}{\sqrt{3}} (\sqrt{3} y) = \frac{3}{\sqrt{3}} \cdot \frac{1}{x}$

Step 2.7.1.1.2.3

Rewrite the expression.

$y = \frac{3}{\sqrt{3}} \cdot \frac{1}{x}$

Step 2.7.2

Simplify the right side.

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Step 2.7.2.1

Simplify $\frac{3}{\sqrt{3}} \cdot \frac{1}{x}$ .

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Step 2.7.2.1.1

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

$y = \frac{3}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \frac{1}{x}$

Step 2.7.2.1.2

Combine and simplify the denominator.

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Step 2.7.2.1.2.1

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

$y = \frac{3 \sqrt{3}}{\sqrt{3} \sqrt{3}} \cdot \frac{1}{x}$

Step 2.7.2.1.2.2

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

$y = \frac{3 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \cdot \frac{1}{x}$

Step 2.7.2.1.2.3

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

$y = \frac{3 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \cdot \frac{1}{x}$

Step 2.7.2.1.2.4

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

$y = \frac{3 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} \cdot \frac{1}{x}$

Step 2.7.2.1.2.5

Add $1$ and $1$ .

$y = \frac{3 \sqrt{3}}{{\sqrt{3}}^{2}} \cdot \frac{1}{x}$

Step 2.7.2.1.2.6

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

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Step 2.7.2.1.2.6.1

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

$y = \frac{3 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \cdot \frac{1}{x}$

Step 2.7.2.1.2.6.2

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

$y = \frac{3 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \cdot \frac{1}{x}$

Step 2.7.2.1.2.6.3

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

$y = \frac{3 \sqrt{3}}{3^{\frac{2}{2}}} \cdot \frac{1}{x}$

Step 2.7.2.1.2.6.4

Cancel the common factor of $2$ .

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Step 2.7.2.1.2.6.4.1

Cancel the common factor.

$y = \frac{3 \sqrt{3}}{3^{\frac{2}{2}}} \cdot \frac{1}{x}$

Step 2.7.2.1.2.6.4.2

Rewrite the expression.

$y = \frac{3 \sqrt{3}}{3^{1}} \cdot \frac{1}{x}$

Step 2.7.2.1.2.6.5

Evaluate the exponent.

$y = \frac{3 \sqrt{3}}{3} \cdot \frac{1}{x}$

Step 2.7.2.1.3

Cancel the common factor of $3$ .

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Step 2.7.2.1.3.1

Cancel the common factor.

$y = \frac{3 \sqrt{3}}{3} \cdot \frac{1}{x}$

Step 2.7.2.1.3.2

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

$y = \sqrt{3} \frac{1}{x}$

Step 2.7.2.1.4

Combine $\sqrt{3}$ and $\frac{1}{x}$ .

$y = \frac{\sqrt{3}}{x}$

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \frac{\sqrt{3}}{x}$

Step 4

Verify if $f^{- 1} (x) = \frac{\sqrt{3}}{x}$ is the inverse of $f (x) = \cot (\arctan (\frac{x}{\sqrt{3}}))$ .

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Step 4.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 4.2

Evaluate $f^{- 1} (f (x))$ .

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Step 4.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 4.2.2

Evaluate $f^{- 1} (\cot (\arctan (\frac{x}{\sqrt{3}})))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\cot (\arctan (\frac{x}{\sqrt{3}}))) = \frac{\sqrt{3}}{\cot (\arctan (\frac{x}{\sqrt{3}}))}$

Step 4.2.3

Separate fractions.

$f^{- 1} (\cot (\arctan (\frac{x}{\sqrt{3}}))) = \frac{\sqrt{3}}{1} \cdot \frac{1}{\cot (\arctan (\frac{x}{\sqrt{3}}))}$

Step 4.2.4

Rewrite $\cot (\arctan (\frac{x}{\sqrt{3}}))$ in terms of sines and cosines.

$f^{- 1} (\cot (\arctan (\frac{x}{\sqrt{3}}))) = \frac{\sqrt{3}}{1} \cdot \frac{1}{\frac{\cos (\arctan (\frac{x}{\sqrt{3}}))}{\sin (\arctan (\frac{x}{\sqrt{3}}))}}$

Step 4.2.5

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (\arctan (\frac{x}{\sqrt{3}}))}{\sin (\arctan (\frac{x}{\sqrt{3}}))}$ .

$f^{- 1} (\cot (\arctan (\frac{x}{\sqrt{3}}))) = \frac{\sqrt{3}}{1} \cdot \frac{\sin (\arctan (\frac{x}{\sqrt{3}}))}{\cos (\arctan (\frac{x}{\sqrt{3}}))}$

Step 4.2.6

Convert from $\frac{\sin (\arctan (\frac{x}{\sqrt{3}}))}{\cos (\arctan (\frac{x}{\sqrt{3}}))}$ to $\tan (\arctan (\frac{x}{\sqrt{3}}))$ .

$f^{- 1} (\cot (\arctan (\frac{x}{\sqrt{3}}))) = \frac{\sqrt{3}}{1} \cdot \tan (\arctan (\frac{x}{\sqrt{3}}))$

Step 4.2.7

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

$f^{- 1} (\cot (\arctan (\frac{x}{\sqrt{3}}))) = \sqrt{3} \tan (\arctan (\frac{x}{\sqrt{3}}))$

Step 4.2.8

The functions tangent and arctangent are inverses.

$f^{- 1} (\cot (\arctan (\frac{x}{\sqrt{3}}))) = \sqrt{3} (\frac{x}{\sqrt{3}})$

Step 4.2.9

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

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Step 4.2.9.1

Cancel the common factor.

$f^{- 1} (\cot (\arctan (\frac{x}{\sqrt{3}}))) = \sqrt{3} (\frac{x}{\sqrt{3}})$

Step 4.2.9.2

Rewrite the expression.

$f^{- 1} (\cot (\arctan (\frac{x}{\sqrt{3}}))) = x$

Step 4.3

Evaluate $f (f^{- 1} (x))$ .

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Step 4.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 4.3.2

Evaluate $f (\frac{\sqrt{3}}{x})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 4.3.3

Draw a triangle in the plane with vertices $(1, \frac{\frac{\sqrt{3}}{x}}{\sqrt{3}})$ , $(1, 0)$ , and the origin. Then $\arctan (\frac{\frac{\sqrt{3}}{x}}{\sqrt{3}})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \frac{\frac{\sqrt{3}}{x}}{\sqrt{3}})$ . Therefore, $\cot (\arctan (\frac{\frac{\sqrt{3}}{x}}{\sqrt{3}}))$ is $\frac{1}{\frac{\frac{\sqrt{3}}{x}}{\sqrt{3}}}$ .

$f (\frac{\sqrt{3}}{x}) = \frac{1}{\frac{\frac{\sqrt{3}}{x}}{\sqrt{3}}}$

Step 4.3.4

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{\sqrt{3}}{x}) = 1 (\frac{\sqrt{3}}{\frac{\sqrt{3}}{x}})$

Step 4.3.5

Multiply $\frac{\sqrt{3}}{\frac{\sqrt{3}}{x}}$ by $1$ .

$f (\frac{\sqrt{3}}{x}) = \frac{\sqrt{3}}{\frac{\sqrt{3}}{x}}$

Step 4.3.6

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{\sqrt{3}}{x}) = \sqrt{3} (\frac{x}{\sqrt{3}})$

Step 4.3.7

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

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Step 4.3.7.1

Cancel the common factor.

$f (\frac{\sqrt{3}}{x}) = \sqrt{3} (\frac{x}{\sqrt{3}})$

Step 4.3.7.2

Rewrite the expression.

$f (\frac{\sqrt{3}}{x}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt{3}}{x}$ is the inverse of $f (x) = \cot (\arctan (\frac{x}{\sqrt{3}}))$ .

$f^{- 1} (x) = \frac{\sqrt{3}}{x}$