Trigonometry Examples

$\frac{\sin (x) \sqrt{2}}{2}$

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

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

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

Step 2.2

Multiply both sides of the equation by $\frac{2}{\sqrt{2}}$ .

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

Step 2.3

Simplify both sides of the equation.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Simplify $\frac{2}{\sqrt{2}} \cdot \frac{\sin (y) \sqrt{2}}{2}$ .

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Step 2.3.1.1.1

Cancel the common factor of $2$ .

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Step 2.3.1.1.1.1

Cancel the common factor.

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

Step 2.3.1.1.1.2

Rewrite the expression.

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

Step 2.3.1.1.2

Cancel the common factor of $\sqrt{2}$ .

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Step 2.3.1.1.2.1

Factor $\sqrt{2}$ out of $\sin (y) \sqrt{2}$ .

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

Step 2.3.1.1.2.2

Cancel the common factor.

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

Step 2.3.1.1.2.3

Rewrite the expression.

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

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\frac{2}{\sqrt{2}} x$ .

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Step 2.3.2.1.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 2.3.2.1.2

Combine and simplify the denominator.

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Step 2.3.2.1.2.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 2.3.2.1.2.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.3.2.1.2.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.3.2.1.2.4

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

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

Step 2.3.2.1.2.5

Add $1$ and $1$ .

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

Step 2.3.2.1.2.6

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

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Step 2.3.2.1.2.6.1

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

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

Step 2.3.2.1.2.6.2

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

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

Step 2.3.2.1.2.6.3

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

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

Step 2.3.2.1.2.6.4

Cancel the common factor of $2$ .

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Step 2.3.2.1.2.6.4.1

Cancel the common factor.

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

Step 2.3.2.1.2.6.4.2

Rewrite the expression.

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

Step 2.3.2.1.2.6.5

Evaluate the exponent.

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

Step 2.3.2.1.3

Cancel the common factor of $2$ .

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Step 2.3.2.1.3.1

Cancel the common factor.

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

Step 2.3.2.1.3.2

Divide $\sqrt{2}$ by $1$ .

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

Step 2.4

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

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

Step 3

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

$f^{- 1} (x) = \arcsin (\sqrt{2} x)$

Step 4

Verify if $f^{- 1} (x) = \arcsin (\sqrt{2} x)$ is the inverse of $f (x) = \frac{\sin (x) \sqrt{2}}{2}$ .

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

$f^{- 1} (\frac{\sin (x) \sqrt{2}}{2}) = \arcsin (\sqrt{2} (\frac{\sin (x) \sqrt{2}}{2}))$

Step 4.2.3

Multiply $\sqrt{2} (\frac{\sin (x) \sqrt{2}}{2})$ .

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Step 4.2.3.1

Combine $\sqrt{2}$ and $\frac{\sin (x) \sqrt{2}}{2}$ .

$f^{- 1} (\frac{\sin (x) \sqrt{2}}{2}) = \arcsin (\frac{\sqrt{2} (\sin (x) \sqrt{2})}{2})$

Step 4.2.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$f^{- 1} (\frac{\sin (x) \sqrt{2}}{2}) = \arcsin (\frac{\sqrt{2} \sqrt{2} \sin (x)}{2})$

Step 4.2.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$f^{- 1} (\frac{\sin (x) \sqrt{2}}{2}) = \arcsin (\frac{\sqrt{2} \sqrt{2} \sin (x)}{2})$

Step 4.2.3.4

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

$f^{- 1} (\frac{\sin (x) \sqrt{2}}{2}) = \arcsin (\frac{{\sqrt{2}}^{1 + 1} \sin (x)}{2})$

Step 4.2.3.5

Add $1$ and $1$ .

$f^{- 1} (\frac{\sin (x) \sqrt{2}}{2}) = \arcsin (\frac{{\sqrt{2}}^{2} \sin (x)}{2})$

Step 4.2.4

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

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Step 4.2.4.1

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

$f^{- 1} (\frac{\sin (x) \sqrt{2}}{2}) = \arcsin (\frac{{(2^{\frac{1}{2}})}^{2} \sin (x)}{2})$

Step 4.2.4.2

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

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

Step 4.2.4.3

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

$f^{- 1} (\frac{\sin (x) \sqrt{2}}{2}) = \arcsin (\frac{2^{\frac{2}{2}} \sin (x)}{2})$

Step 4.2.4.4

Cancel the common factor of $2$ .

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Step 4.2.4.4.1

Cancel the common factor.

$f^{- 1} (\frac{\sin (x) \sqrt{2}}{2}) = \arcsin (\frac{2^{\frac{2}{2}} \sin (x)}{2})$

Step 4.2.4.4.2

Rewrite the expression.

$f^{- 1} (\frac{\sin (x) \sqrt{2}}{2}) = \arcsin (\frac{2 \sin (x)}{2})$

Step 4.2.4.5

Evaluate the exponent.

$f^{- 1} (\frac{\sin (x) \sqrt{2}}{2}) = \arcsin (\frac{2 \sin (x)}{2})$

Step 4.2.5

Cancel the common factor of $2$ .

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Step 4.2.5.1

Cancel the common factor.

$f^{- 1} (\frac{\sin (x) \sqrt{2}}{2}) = \arcsin (\frac{2 \sin (x)}{2})$

Step 4.2.5.2

Divide $\sin (x)$ by $1$ .

$f^{- 1} (\frac{\sin (x) \sqrt{2}}{2}) = \arcsin (\sin (x))$

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

$f (\arcsin (\sqrt{2} x)) = \frac{\sin (\arcsin (\sqrt{2} x)) \sqrt{2}}{2}$

Step 4.3.3

Simplify the numerator.

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Step 4.3.3.1

The functions sine and arcsine are inverses.

$f (\arcsin (\sqrt{2} x)) = \frac{\sqrt{2} x \sqrt{2}}{2}$

Step 4.3.3.2

Combine exponents.

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Step 4.3.3.2.1

Raise $\sqrt{2}$ to the power of $1$ .

$f (\arcsin (\sqrt{2} x)) = \frac{x (\sqrt{2} \sqrt{2})}{2}$

Step 4.3.3.2.2

Raise $\sqrt{2}$ to the power of $1$ .

$f (\arcsin (\sqrt{2} x)) = \frac{x (\sqrt{2} \sqrt{2})}{2}$

Step 4.3.3.2.3

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

$f (\arcsin (\sqrt{2} x)) = \frac{x {\sqrt{2}}^{1 + 1}}{2}$

Step 4.3.3.2.4

Add $1$ and $1$ .

$f (\arcsin (\sqrt{2} x)) = \frac{x {\sqrt{2}}^{2}}{2}$

Step 4.3.3.3

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

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Step 4.3.3.3.1

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

$f (\arcsin (\sqrt{2} x)) = \frac{x {(2^{\frac{1}{2}})}^{2}}{2}$

Step 4.3.3.3.2

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

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

Step 4.3.3.3.3

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

$f (\arcsin (\sqrt{2} x)) = \frac{x \cdot 2^{\frac{2}{2}}}{2}$

Step 4.3.3.3.4

Cancel the common factor of $2$ .

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Step 4.3.3.3.4.1

Cancel the common factor.

$f (\arcsin (\sqrt{2} x)) = \frac{x \cdot 2^{\frac{2}{2}}}{2}$

Step 4.3.3.3.4.2

Rewrite the expression.

$f (\arcsin (\sqrt{2} x)) = \frac{x \cdot 2}{2}$

Step 4.3.3.3.5

Evaluate the exponent.

$f (\arcsin (\sqrt{2} x)) = \frac{x \cdot 2}{2}$

Step 4.3.4

Cancel the common factor of $2$ .

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Step 4.3.4.1

Cancel the common factor.

$f (\arcsin (\sqrt{2} x)) = \frac{x \cdot 2}{2}$

Step 4.3.4.2

Divide $x$ by $1$ .

$f (\arcsin (\sqrt{2} x)) = x$

Step 4.4

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

$f^{- 1} (x) = \arcsin (\sqrt{2} x)$