Trigonometry Examples

$\sin (x) \sqrt{\frac{3}{2}}$

Step 1

Interchange the variables.

$x = \sin (y) \sqrt{\frac{3}{2}}$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $\sin (y) \sqrt{\frac{3}{2}} = x$ .

$\sin (y) \sqrt{\frac{3}{2}} = x$

Step 2.2

Divide each term in $\sin (y) \sqrt{\frac{3}{2}} = x$ by $\sqrt{\frac{3}{2}}$ and simplify.

Tap for more steps...

Step 2.2.1

Divide each term in $\sin (y) \sqrt{\frac{3}{2}} = x$ by $\sqrt{\frac{3}{2}}$ .

$\frac{\sin (y) \sqrt{\frac{3}{2}}}{\sqrt{\frac{3}{2}}} = \frac{x}{\sqrt{\frac{3}{2}}}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Cancel the common factor of $\sqrt{\frac{3}{2}}$ .

Tap for more steps...

Step 2.2.2.1.1

Cancel the common factor.

$\frac{\sin (y) \sqrt{\frac{3}{2}}}{\sqrt{\frac{3}{2}}} = \frac{x}{\sqrt{\frac{3}{2}}}$

Step 2.2.2.1.2

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

$\sin (y) = \frac{x}{\sqrt{\frac{3}{2}}}$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Simplify the denominator.

Tap for more steps...

Step 2.2.3.1.1

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

$\sin (y) = \frac{x}{\frac{\sqrt{3}}{\sqrt{2}}}$

Step 2.2.3.1.2

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 2.2.3.1.3

Combine and simplify the denominator.

Tap for more steps...

Step 2.2.3.1.3.1

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (y) = \frac{x}{\frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}}}$

Step 2.2.3.1.3.2

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

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

Step 2.2.3.1.3.3

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

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

Step 2.2.3.1.3.4

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

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

Step 2.2.3.1.3.5

Add $1$ and $1$ .

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

Step 2.2.3.1.3.6

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

Tap for more steps...

Step 2.2.3.1.3.6.1

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

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

Step 2.2.3.1.3.6.2

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

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

Step 2.2.3.1.3.6.3

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

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

Step 2.2.3.1.3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.3.1.3.6.4.1

Cancel the common factor.

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

Step 2.2.3.1.3.6.4.2

Rewrite the expression.

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

Step 2.2.3.1.3.6.5

Evaluate the exponent.

$\sin (y) = \frac{x}{\frac{\sqrt{3} \sqrt{2}}{2}}$

Step 2.2.3.1.4

Simplify the numerator.

Tap for more steps...

Step 2.2.3.1.4.1

Combine using the product rule for radicals.

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

Step 2.2.3.1.4.2

Multiply $3$ by $2$ .

$\sin (y) = \frac{x}{\frac{\sqrt{6}}{2}}$

Step 2.2.3.2

Multiply the numerator by the reciprocal of the denominator.

$\sin (y) = x \frac{2}{\sqrt{6}}$

Step 2.2.3.3

Multiply $\frac{2}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\sin (y) = x (\frac{2}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}})$

Step 2.2.3.4

Combine and simplify the denominator.

Tap for more steps...

Step 2.2.3.4.1

Multiply $\frac{2}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\sin (y) = x \frac{2 \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 2.2.3.4.2

Raise $\sqrt{6}$ to the power of $1$ .

$\sin (y) = x \frac{2 \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Step 2.2.3.4.3

Raise $\sqrt{6}$ to the power of $1$ .

$\sin (y) = x \frac{2 \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

Step 2.2.3.4.4

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

$\sin (y) = x \frac{2 \sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Step 2.2.3.4.5

Add $1$ and $1$ .

$\sin (y) = x \frac{2 \sqrt{6}}{{\sqrt{6}}^{2}}$

Step 2.2.3.4.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Tap for more steps...

Step 2.2.3.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\sin (y) = x \frac{2 \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

Step 2.2.3.4.6.2

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

$\sin (y) = x \frac{2 \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Step 2.2.3.4.6.3

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

$\sin (y) = x \frac{2 \sqrt{6}}{6^{\frac{2}{2}}}$

Step 2.2.3.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.3.4.6.4.1

Cancel the common factor.

$\sin (y) = x \frac{2 \sqrt{6}}{6^{\frac{2}{2}}}$

Step 2.2.3.4.6.4.2

Rewrite the expression.

$\sin (y) = x \frac{2 \sqrt{6}}{6^{1}}$

Step 2.2.3.4.6.5

Evaluate the exponent.

$\sin (y) = x \frac{2 \sqrt{6}}{6}$

Step 2.2.3.5

Cancel the common factor of $2$ and $6$ .

Tap for more steps...

Step 2.2.3.5.1

Factor $2$ out of $2 \sqrt{6}$ .

$\sin (y) = x \frac{2 (\sqrt{6})}{6}$

Step 2.2.3.5.2

Cancel the common factors.

Tap for more steps...

Step 2.2.3.5.2.1

Factor $2$ out of $6$ .

$\sin (y) = x \frac{2 \sqrt{6}}{2 \cdot 3}$

Step 2.2.3.5.2.2

Cancel the common factor.

$\sin (y) = x \frac{2 \sqrt{6}}{2 \cdot 3}$

Step 2.2.3.5.2.3

Rewrite the expression.

$\sin (y) = x \frac{\sqrt{6}}{3}$

Step 2.2.3.6

Combine $x$ and $\frac{\sqrt{6}}{3}$ .

$\sin (y) = \frac{x \sqrt{6}}{3}$

Step 2.3

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

$y = \arcsin (\frac{x \sqrt{6}}{3})$

Step 3

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

$f^{- 1} (x) = \arcsin (\frac{x \sqrt{6}}{3})$

Step 4

Verify if $f^{- 1} (x) = \arcsin (\frac{x \sqrt{6}}{3})$ is the inverse of $f (x) = \sin (x) \sqrt{\frac{3}{2}}$ .

Tap for more steps...

Step 4.1

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

Step 4.2

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

Tap for more steps...

Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (\sin (x) \sqrt{\frac{3}{2}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sin (x) \sqrt{\frac{3}{2}}) = \arcsin (\frac{(\sin (x) \sqrt{\frac{3}{2}}) \sqrt{6}}{3})$

Step 4.2.3

Simplify the numerator.

Tap for more steps...

Step 4.2.3.1

Combine using the product rule for radicals.

$f^{- 1} (\sin (x) \sqrt{\frac{3}{2}}) = \arcsin (\frac{\sin (x) \sqrt{6 (\frac{3}{2})}}{3})$

Step 4.2.3.2

Combine $6$ and $\frac{3}{2}$ .

$f^{- 1} (\sin (x) \sqrt{\frac{3}{2}}) = \arcsin (\frac{\sin (x) \sqrt{\frac{6 \cdot 3}{2}}}{3})$

Step 4.2.4

Multiply $6$ by $3$ .

$f^{- 1} (\sin (x) \sqrt{\frac{3}{2}}) = \arcsin (\frac{\sin (x) \sqrt{\frac{18}{2}}}{3})$

Step 4.2.5

Simplify the numerator.

Tap for more steps...

Step 4.2.5.1

Divide $18$ by $2$ .

$f^{- 1} (\sin (x) \sqrt{\frac{3}{2}}) = \arcsin (\frac{\sin (x) \sqrt{9}}{3})$

Step 4.2.5.2

Rewrite $9$ as $3^{2}$ .

$f^{- 1} (\sin (x) \sqrt{\frac{3}{2}}) = \arcsin (\frac{\sin (x) \sqrt{3^{2}}}{3})$

Step 4.2.5.3

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

$f^{- 1} (\sin (x) \sqrt{\frac{3}{2}}) = \arcsin (\frac{\sin (x) \cdot 3}{3})$

Step 4.2.6

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.6.1

Cancel the common factor.

$f^{- 1} (\sin (x) \sqrt{\frac{3}{2}}) = \arcsin (\frac{\sin (x) \cdot 3}{3})$

Step 4.2.6.2

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

$f^{- 1} (\sin (x) \sqrt{\frac{3}{2}}) = \arcsin (\sin (x))$

Step 4.3

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

Tap for more steps...

Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\arcsin (\frac{x \sqrt{6}}{3}))$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \sin (\arcsin (\frac{x \sqrt{6}}{3})) \sqrt{\frac{3}{2}}$

Step 4.3.3

The functions sine and arcsine are inverses.

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \sqrt{6}}{3} \cdot \sqrt{\frac{3}{2}}$

Step 4.3.4

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

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \sqrt{6}}{3} \cdot \frac{\sqrt{3}}{\sqrt{2}}$

Step 4.3.5

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \sqrt{6}}{3} \cdot (\frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 4.3.6

Combine and simplify the denominator.

Tap for more steps...

Step 4.3.6.1

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \sqrt{6}}{3} \cdot \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 4.3.6.2

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

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \sqrt{6}}{3} \cdot \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 4.3.6.3

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

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \sqrt{6}}{3} \cdot \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 4.3.6.4

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

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \sqrt{6}}{3} \cdot \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 4.3.6.5

Add $1$ and $1$ .

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \sqrt{6}}{3} \cdot \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 4.3.6.6

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

Tap for more steps...

Step 4.3.6.6.1

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

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \sqrt{6}}{3} \cdot \frac{\sqrt{3} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 4.3.6.6.2

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

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \sqrt{6}}{3} \cdot \frac{\sqrt{3} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 4.3.6.6.3

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

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \sqrt{6}}{3} \cdot \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.3.6.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.6.6.4.1

Cancel the common factor.

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \sqrt{6}}{3} \cdot \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.3.6.6.4.2

Rewrite the expression.

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \sqrt{6}}{3} \cdot \frac{\sqrt{3} \sqrt{2}}{2}$

Step 4.3.6.6.5

Evaluate the exponent.

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \sqrt{6}}{3} \cdot \frac{\sqrt{3} \sqrt{2}}{2}$

Step 4.3.7

Simplify the numerator.

Tap for more steps...

Step 4.3.7.1

Combine using the product rule for radicals.

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \sqrt{6}}{3} \cdot \frac{\sqrt{3 \cdot 2}}{2}$

Step 4.3.7.2

Multiply $3$ by $2$ .

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \sqrt{6}}{3} \cdot \frac{\sqrt{6}}{2}$

Step 4.3.8

Multiply $\frac{x \sqrt{6}}{3} \cdot \frac{\sqrt{6}}{2}$ .

Tap for more steps...

Step 4.3.8.1

Multiply $\frac{x \sqrt{6}}{3}$ by $\frac{\sqrt{6}}{2}$ .

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \sqrt{6} \sqrt{6}}{3 \cdot 2}$

Step 4.3.8.2

Raise $\sqrt{6}$ to the power of $1$ .

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x (\sqrt{6} \sqrt{6})}{3 \cdot 2}$

Step 4.3.8.3

Raise $\sqrt{6}$ to the power of $1$ .

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x (\sqrt{6} \sqrt{6})}{3 \cdot 2}$

Step 4.3.8.4

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

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x {\sqrt{6}}^{1 + 1}}{3 \cdot 2}$

Step 4.3.8.5

Add $1$ and $1$ .

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x {\sqrt{6}}^{2}}{3 \cdot 2}$

Step 4.3.8.6

Multiply $3$ by $2$ .

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x {\sqrt{6}}^{2}}{6}$

Step 4.3.9

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Tap for more steps...

Step 4.3.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x {(6^{\frac{1}{2}})}^{2}}{6}$

Step 4.3.9.2

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

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \cdot 6^{\frac{1}{2} \cdot 2}}{6}$

Step 4.3.9.3

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

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \cdot 6^{\frac{2}{2}}}{6}$

Step 4.3.9.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.9.4.1

Cancel the common factor.

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \cdot 6^{\frac{2}{2}}}{6}$

Step 4.3.9.4.2

Rewrite the expression.

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \cdot 6}{6}$

Step 4.3.9.5

Evaluate the exponent.

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \cdot 6}{6}$

Step 4.3.10

Cancel the common factor of $6$ .

Tap for more steps...

Step 4.3.10.1

Cancel the common factor.

$f (\arcsin (\frac{x \sqrt{6}}{3})) = \frac{x \cdot 6}{6}$

Step 4.3.10.2

Divide $x$ by $1$ .

$f (\arcsin (\frac{x \sqrt{6}}{3})) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \arcsin (\frac{x \sqrt{6}}{3})$ is the inverse of $f (x) = \sin (x) \sqrt{\frac{3}{2}}$ .

$f^{- 1} (x) = \arcsin (\frac{x \sqrt{6}}{3})$