Trigonometry Examples

cot (arctan (\sqrt{\frac{2}{x}}))

$\cot (\arctan (\sqrt{\frac{2}{x}}))$

Step 1

Interchange the variables.

x = cot (arctan (\sqrt{\frac{2}{y}}))

$x = \cot (\arctan (\sqrt{\frac{2}{y}}))$

Step 2

Solve for

y

$y$ .

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Step 2.1

Rewrite the equation as

cot (arctan (\sqrt{\frac{2}{y}})) = x

$\cot (\arctan (\sqrt{\frac{2}{y}})) = x$ .

cot (arctan (\sqrt{\frac{2}{y}})) = x

$\cot (\arctan (\sqrt{\frac{2}{y}})) = x$

Step 2.2

Take the inverse cotangent of both sides of the equation to extract

arctan (\sqrt{\frac{2}{y}})

$\arctan (\sqrt{\frac{2}{y}})$ from inside the cotangent.

arctan (\sqrt{\frac{2}{y}}) = arccot (x)

$\arctan (\sqrt{\frac{2}{y}}) = arccot (x)$

Step 2.3

Take the inverse arctangent of both sides of the equation to extract

y

$y$ from inside the arctangent.

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

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

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify

\sqrt{\frac{2}{y}}

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

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Step 2.4.1.1

Rewrite

\sqrt{\frac{2}{y}}

$\sqrt{\frac{2}{y}}$ as

\frac{\sqrt{2}}{\sqrt{y}}

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

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

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

Step 2.4.1.2

Multiply

\frac{\sqrt{2}}{\sqrt{y}}

$\frac{\sqrt{2}}{\sqrt{y}}$ by

\frac{\sqrt{y}}{\sqrt{y}}

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

\frac{\sqrt{2}}{\sqrt{y}} \cdot \frac{\sqrt{y}}{\sqrt{y}} = tan (arccot (x))

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

Step 2.4.1.3

Combine and simplify the denominator.

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Step 2.4.1.3.1

Multiply

\frac{\sqrt{2}}{\sqrt{y}}

$\frac{\sqrt{2}}{\sqrt{y}}$ by

\frac{\sqrt{y}}{\sqrt{y}}

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

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

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

Step 2.4.1.3.2

Raise

\sqrt{y}

$\sqrt{y}$ to the power of

1

$1$ .

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

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

Step 2.4.1.3.3

Raise

\sqrt{y}

$\sqrt{y}$ to the power of

1

$1$ .

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

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

Step 2.4.1.3.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 2.4.1.3.5

Add

1

$1$ and

1

$1$ .

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

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

Step 2.4.1.3.6

Rewrite

{\sqrt{y}}^{2}

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

y

$y$ .

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Step 2.4.1.3.6.1

Use

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

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

\sqrt{y}

$\sqrt{y}$ as

y^{\frac{1}{2}}

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

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

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

Step 2.4.1.3.6.2

Apply the power rule and multiply exponents,

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

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

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

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

Step 2.4.1.3.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 2.4.1.3.6.4

Cancel the common factor of

2

$2$ .

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Step 2.4.1.3.6.4.1

Cancel the common factor.

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

Step 2.4.1.3.6.4.2

Rewrite the expression.

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

Step 2.4.1.3.6.5

Simplify.

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

Step 2.4.1.4

Combine using the product rule for radicals.

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

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

$\frac{1}{x}$ .

$\frac{\sqrt{2 y}}{y} = \frac{1}{x}$

Step 2.6

Cross multiply.

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Step 2.6.1

Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.

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

Step 2.6.2

Simplify the left side.

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Step 2.6.2.1

Multiply

$y$ by

$1$ .

$y = \sqrt{2 y} \cdot (x)$

Step 2.6.3

Simplify the right side.

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Step 2.6.3.1

Multiply

$\sqrt{2 y}$ by

$x$ .

$y = \sqrt{2 y} x$

Step 2.7

Rewrite the equation as

$\sqrt{2 y} x = y$ .

$\sqrt{2 y} x = y$

Step 2.8

To remove the radical on the left side of the equation, square both sides of the equation.

${(\sqrt{2 y} x)}^{2} = y^{2}$

Step 2.9

Simplify each side of the equation.

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Step 2.9.1

Use

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

$\sqrt{2 y}$ as

${(2 y)}^{\frac{1}{2}}$ .

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

Step 2.9.2

Simplify the left side.

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Step 2.9.2.1

Simplify

${({(2 y)}^{\frac{1}{2}} x)}^{2}$ .

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Step 2.9.2.1.1

Apply the product rule to

$2 y$ .

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

Step 2.9.2.1.2

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.9.2.1.2.1

Apply the product rule to

$2^{\frac{1}{2}} y^{\frac{1}{2}} x$ .

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

Step 2.9.2.1.2.2

Apply the product rule to

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

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

Step 2.9.2.1.3

Multiply the exponents in

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

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Step 2.9.2.1.3.1

Apply the power rule and multiply exponents,

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

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

Step 2.9.2.1.3.2

Cancel the common factor of

$2$ .

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Step 2.9.2.1.3.2.1

Cancel the common factor.

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

Step 2.9.2.1.3.2.2

Rewrite the expression.

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

Step 2.9.2.1.4

Evaluate the exponent.

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

Step 2.9.2.1.5

Multiply the exponents in

${(y^{\frac{1}{2}})}^{2}$ .

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Step 2.9.2.1.5.1

Apply the power rule and multiply exponents,

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

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

Step 2.9.2.1.5.2

Cancel the common factor of

$2$ .

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Step 2.9.2.1.5.2.1

Cancel the common factor.

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

Step 2.9.2.1.5.2.2

Rewrite the expression.

$2 y^{1} x^{2} = y^{2}$

Step 2.9.2.1.6

Simplify.

$2 y x^{2} = y^{2}$

Step 2.10

Solve for

$y$ .

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Step 2.10.1

Subtract

$y^{2}$ from both sides of the equation.

$2 y x^{2} - y^{2} = 0$

Step 2.10.2

Factor

$y$ out of

$2 y x^{2} - y^{2}$ .

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Step 2.10.2.1

Factor

$y$ out of

$2 y x^{2}$ .

$y (2 x^{2}) - y^{2} = 0$

Step 2.10.2.2

Factor

$y$ out of

$- y^{2}$ .

$y (2 x^{2}) + y (- y) = 0$

Step 2.10.2.3

Factor

$y$ out of

$y (2 x^{2}) + y (- y)$ .

$y (2 x^{2} - y) = 0$

Step 2.10.3

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$y = 0$

$2 x^{2} - y = 0$

Step 2.10.4

Set

$y$ equal to

$0$ .

$y = 0$

Step 2.10.5

Set

$2 x^{2} - y$ equal to

$0$ and solve for

$y$ .

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Step 2.10.5.1

Set

$2 x^{2} - y$ equal to

$0$ .

$2 x^{2} - y = 0$

Step 2.10.5.2

Solve

$2 x^{2} - y = 0$ for

$y$ .

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Step 2.10.5.2.1

Subtract

$2 x^{2}$ from both sides of the equation.

$- y = - 2 x^{2}$

Step 2.10.5.2.2

Divide each term in

$- y = - 2 x^{2}$ by

$- 1$ and simplify.

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Step 2.10.5.2.2.1

Divide each term in

$- y = - 2 x^{2}$ by

$- 1$ .

$\frac{- y}{- 1} = \frac{- 2 x^{2}}{- 1}$

Step 2.10.5.2.2.2

Simplify the left side.

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Step 2.10.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 2 x^{2}}{- 1}$

Step 2.10.5.2.2.2.2

Divide

$y$ by

$1$ .

$y = \frac{- 2 x^{2}}{- 1}$

Step 2.10.5.2.2.3

Simplify the right side.

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Step 2.10.5.2.2.3.1

Move the negative one from the denominator of

$\frac{- 2 x^{2}}{- 1}$ .

$y = - 1 \cdot (- 2 x^{2})$

Step 2.10.5.2.2.3.2

Rewrite

$- 1 \cdot (- 2 x^{2})$ as

$- (- 2 x^{2})$ .

$y = - (- 2 x^{2})$

Step 2.10.5.2.2.3.3

Multiply

$- 2$ by

$- 1$ .

$y = 2 x^{2}$

Step 2.10.6

The final solution is all the values that make

$y (2 x^{2} - y) = 0$ true.

$y = 0$

$y = 2 x^{2}$

$y = 0$

$y = 2 x^{2}$

$y = 0$

$y = 2 x^{2}$

Step 3

Replace

$y$ with

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

$f^{- 1} (x) = 0, 2 x^{2}$

Step 4

Verify if

$f^{- 1} (x) = 0, 2 x^{2}$ is the inverse of

$f (x) = \cot (\arctan (\sqrt{\frac{2}{x}}))$ .

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Step 4.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of

$f (x) = \cot (\arctan (\sqrt{\frac{2}{x}}))$ and

$f^{- 1} (x) = 0, 2 x^{2}$ and compare them.

Step 4.2

Find the domain of

$0$ .

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Step 4.2.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

$(- \infty, \infty)$

Step 4.3

Since the domain of

$f^{- 1} (x) = 0, 2 x^{2}$ is not equal to the range of

$f (x) = \cot (\arctan (\sqrt{\frac{2}{x}}))$ , then

$f^{- 1} (x) = 0, 2 x^{2}$ is not an inverse of

$f (x) = \cot (\arctan (\sqrt{\frac{2}{x}}))$ .

There is no inverse

Step 5