Trigonometry Examples

$\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}$

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

Simplify $\frac{1}{2} \cdot {\arctan (\sqrt{y})}^{- \frac{1}{2}}$ .

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Step 2.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.1.2

Combine.

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

Step 2.1.3

Multiply $1$ by $1$ .

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

Step 2.2

Rewrite the equation as $\frac{1}{2 {\arctan (\sqrt{y})}^{\frac{1}{2}}} = x$ .

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

Step 2.3

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

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

Step 2.4

Find the LCD of the terms in the equation.

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Step 2.4.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2 {\arctan (y^{\frac{1}{2}})}^{\frac{1}{2}}, 1$

Step 2.4.2

The LCM of one and any expression is the expression.

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

Step 2.5

Multiply each term in $\frac{1}{2 {\arctan (y^{\frac{1}{2}})}^{\frac{1}{2}}} = x$ by $2 {\arctan (y^{\frac{1}{2}})}^{\frac{1}{2}}$ to eliminate the fractions.

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Step 2.5.1

Multiply each term in $\frac{1}{2 {\arctan (y^{\frac{1}{2}})}^{\frac{1}{2}}} = x$ by $2 {\arctan (y^{\frac{1}{2}})}^{\frac{1}{2}}$ .

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

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Rewrite using the commutative property of multiplication.

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

Step 2.5.2.2

Cancel the common factor of $2$ .

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Step 2.5.2.2.1

Cancel the common factor.

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

Step 2.5.2.2.2

Rewrite the expression.

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

Step 2.5.2.3

Cancel the common factor of ${\arctan (y^{\frac{1}{2}})}^{\frac{1}{2}}$ .

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Step 2.5.2.3.1

Cancel the common factor.

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

Step 2.5.2.3.2

Rewrite the expression.

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

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Rewrite using the commutative property of multiplication.

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

Step 2.6

Solve the equation.

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Step 2.6.1

Rewrite the equation as $2 x {\arctan (y^{\frac{1}{2}})}^{\frac{1}{2}} = 1$ .

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

Step 2.6.2

Divide each term in $2 x {\arctan (y^{\frac{1}{2}})}^{\frac{1}{2}} = 1$ by $2 x$ and simplify.

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Step 2.6.2.1

Divide each term in $2 x {\arctan (y^{\frac{1}{2}})}^{\frac{1}{2}} = 1$ by $2 x$ .

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

Step 2.6.2.2

Simplify the left side.

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Step 2.6.2.2.1

Cancel the common factor.

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

Step 2.6.2.2.2

Rewrite the expression.

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

Step 2.6.2.2.3

Cancel the common factor.

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

Step 2.6.2.2.4

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

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

Step 2.6.3

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 2.6.4

Simplify the exponent.

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Step 2.6.4.1

Simplify the left side.

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Step 2.6.4.1.1

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

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Step 2.6.4.1.1.1

Multiply the exponents in ${({\arctan (y^{\frac{1}{2}})}^{\frac{1}{2}})}^{2}$ .

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Step 2.6.4.1.1.1.1

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

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

Step 2.6.4.1.1.1.2

Cancel the common factor of $2$ .

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Step 2.6.4.1.1.1.2.1

Cancel the common factor.

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

Step 2.6.4.1.1.1.2.2

Rewrite the expression.

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

Step 2.6.4.1.1.2

Simplify.

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

Step 2.6.4.2

Simplify the right side.

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Step 2.6.4.2.1

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

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Step 2.6.4.2.1.1

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

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Step 2.6.4.2.1.1.1

Apply the product rule to $\frac{1}{2 x}$ .

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

Step 2.6.4.2.1.1.2

Apply the product rule to $2 x$ .

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

Step 2.6.4.2.1.2

One to any power is one.

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

Step 2.6.4.2.1.3

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

$\arctan (y^{\frac{1}{2}}) = \frac{1}{4 x^{2}}$

Step 2.6.5

Solve for $y$ .

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Step 2.6.5.1

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

$y^{\frac{1}{2}} = \tan (\frac{1}{4 x^{2}})$

Step 2.6.5.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(y^{\frac{1}{2}})}^{2} = \tan^{2} (\frac{1}{4 x^{2}})$

Step 2.6.5.3

Simplify the left side.

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Step 2.6.5.3.1

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

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Step 2.6.5.3.1.1

Multiply the exponents in ${(y^{\frac{1}{2}})}^{2}$ .

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Step 2.6.5.3.1.1.1

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

$y^{\frac{1}{2} \cdot 2} = \tan^{2} (\frac{1}{4 x^{2}})$

Step 2.6.5.3.1.1.2

Cancel the common factor of $2$ .

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Step 2.6.5.3.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{2} \cdot 2} = \tan^{2} (\frac{1}{4 x^{2}})$

Step 2.6.5.3.1.1.2.2

Rewrite the expression.

$y^{1} = \tan^{2} (\frac{1}{4 x^{2}})$

Step 2.6.5.3.1.2

Simplify.

$y = \tan^{2} (\frac{1}{4 x^{2}})$

Step 3

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

$f^{- 1} (x) = \tan^{2} (\frac{1}{4 x^{2}})$

Step 4

Verify if $f^{- 1} (x) = \tan^{2} (\frac{1}{4 x^{2}})$ is the inverse of $f (x) = \frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}$ .

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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 $f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = \tan^{2} (\frac{1}{4 {(\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}})}^{2}})$

Step 4.2.3

Simplify the denominator.

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Step 4.2.3.1

Apply the product rule to $\frac{1}{2} \cdot {\arctan (\sqrt{x})}^{- \frac{1}{2}}$ .

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = \tan^{2} (\frac{1}{4 {(\frac{1}{2})}^{2} \cdot {({\arctan (\sqrt{x})}^{- \frac{1}{2}})}^{2}})$

Step 4.2.3.2

Combine exponents.

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Step 4.2.3.2.1

Rewrite $4$ as $2^{2}$ .

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = \tan^{2} (\frac{1}{2^{2} {(\frac{1}{2})}^{2} \cdot {({\arctan (\sqrt{x})}^{- \frac{1}{2}})}^{2}})$

Step 4.2.3.2.2

Rewrite $\frac{1}{2}$ as $2^{- 1}$ .

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = \tan^{2} (\frac{1}{2^{2} {(2^{- 1})}^{2} \cdot {({\arctan (\sqrt{x})}^{- \frac{1}{2}})}^{2}})$

Step 4.2.3.2.3

Raise $2$ to the power of $1$ .

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = \tan^{2} (\frac{1}{2^{2} {({(2)}^{- 1})}^{2} \cdot {({\arctan (\sqrt{x})}^{- \frac{1}{2}})}^{2}})$

Step 4.2.3.2.4

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

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = \tan^{2} (\frac{1}{2^{2} {(2^{1 \cdot - 1})}^{2} \cdot {({\arctan (\sqrt{x})}^{- \frac{1}{2}})}^{2}})$

Step 4.2.3.2.5

Multiply $- 1$ by $1$ .

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = \tan^{2} (\frac{1}{2^{2} {(2^{- 1})}^{2} \cdot {({\arctan (\sqrt{x})}^{- \frac{1}{2}})}^{2}})$

Step 4.2.3.2.6

Multiply the exponents in ${(2^{- 1})}^{2}$ .

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Step 4.2.3.2.6.1

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

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = \tan^{2} (\frac{1}{2^{2} \cdot 2^{- 1 \cdot 2} \cdot {({\arctan (\sqrt{x})}^{- \frac{1}{2}})}^{2}})$

Step 4.2.3.2.6.2

Multiply $- 1$ by $2$ .

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = \tan^{2} (\frac{1}{2^{2} \cdot 2^{- 2} \cdot {({\arctan (\sqrt{x})}^{- \frac{1}{2}})}^{2}})$

Step 4.2.3.2.7

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

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = \tan^{2} (\frac{1}{2^{2 - 2} \cdot {({\arctan (\sqrt{x})}^{- \frac{1}{2}})}^{2}})$

Step 4.2.3.2.8

Subtract $2$ from $2$ .

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = \tan^{2} (\frac{1}{2^{0} \cdot {({\arctan (\sqrt{x})}^{- \frac{1}{2}})}^{2}})$

Step 4.2.3.3

Anything raised to $0$ is $1$ .

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = \tan^{2} (\frac{1}{1 \cdot {({\arctan (\sqrt{x})}^{- \frac{1}{2}})}^{2}})$

Step 4.2.3.4

Multiply the exponents in ${({\arctan (\sqrt{x})}^{- \frac{1}{2}})}^{2}$ .

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Step 4.2.3.4.1

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

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = \tan^{2} (\frac{1}{1 \cdot {\arctan (\sqrt{x})}^{- \frac{1}{2} \cdot 2}})$

Step 4.2.3.4.2

Cancel the common factor of $2$ .

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Step 4.2.3.4.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = \tan^{2} (\frac{1}{1 \cdot {\arctan (\sqrt{x})}^{\frac{- 1}{2} \cdot 2}})$

Step 4.2.3.4.2.2

Cancel the common factor.

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = \tan^{2} (\frac{1}{1 \cdot {\arctan (\sqrt{x})}^{\frac{- 1}{2} \cdot 2}})$

Step 4.2.3.4.2.3

Rewrite the expression.

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = \tan^{2} (\frac{1}{1 \cdot {\arctan (\sqrt{x})}^{- 1}})$

Step 4.2.3.5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = \tan^{2} (\frac{1}{1 \cdot \frac{1}{\arctan (\sqrt{x})}})$

Step 4.2.3.6

Multiply $\frac{1}{\arctan (\sqrt{x})}$ by $1$ .

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = \tan^{2} (\frac{1}{\frac{1}{\arctan (\sqrt{x})}})$

Step 4.2.4

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = \tan^{2} (1 \arctan (\sqrt{x}))$

Step 4.2.5

Multiply $\arctan (\sqrt{x})$ by $1$ .

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = \tan^{2} (\arctan (\sqrt{x}))$

Step 4.2.6

The functions tangent and arctangent are inverses.

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = {\sqrt{x}}^{2}$

Step 4.2.7

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

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Step 4.2.7.1

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

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = {(x^{\frac{1}{2}})}^{2}$

Step 4.2.7.2

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

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = x^{\frac{1}{2} \cdot 2}$

Step 4.2.7.3

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

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = x^{\frac{2}{2}}$

Step 4.2.7.4

Cancel the common factor of $2$ .

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Step 4.2.7.4.1

Cancel the common factor.

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = x^{\frac{2}{2}}$

Step 4.2.7.4.2

Rewrite the expression.

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = x$

Step 4.2.7.5

Simplify.

$f^{- 1} (\frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}) = x$

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 $f (\tan^{2} (\frac{1}{4 x^{2}}))$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\tan^{2} (\frac{1}{4 x^{2}})) = \frac{1}{2} \cdot {(\arctan (\sqrt{\tan^{2} (\frac{1}{4 x^{2}})}))}^{- \frac{1}{2}}$

Step 4.3.3

Pull terms out from under the radical, assuming positive real numbers.

$f (\tan^{2} (\frac{1}{4 x^{2}})) = \frac{1}{2} \cdot {\arctan (\tan (\frac{1}{4 x^{2}}))}^{- \frac{1}{2}}$

Step 4.3.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f (\tan^{2} (\frac{1}{4 x^{2}})) = \frac{1}{2} \cdot \frac{1}{{\arctan (\tan (\frac{1}{4 x^{2}}))}^{\frac{1}{2}}}$

Step 4.3.5

Combine.

$f (\tan^{2} (\frac{1}{4 x^{2}})) = \frac{1 \cdot 1}{2 {\arctan (\tan (\frac{1}{4 x^{2}}))}^{\frac{1}{2}}}$

Step 4.3.6

Multiply $1$ by $1$ .

$f (\tan^{2} (\frac{1}{4 x^{2}})) = \frac{1}{2 {\arctan (\tan (\frac{1}{4 x^{2}}))}^{\frac{1}{2}}}$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \tan^{2} (\frac{1}{4 x^{2}})$ is the inverse of $f (x) = \frac{1}{2} \cdot {(\arctan (\sqrt{x}))}^{- \frac{1}{2}}$ .

$f^{- 1} (x) = \tan^{2} (\frac{1}{4 x^{2}})$