Trigonometry Examples

$\frac{\sin (30)}{x} = \frac{\sin (60)}{y}$

Step 1

Rewrite the equation as $\frac{\sin (60)}{y} = \frac{\sin (30)}{x}$ .

$\frac{\sin (60)}{y} = \frac{\sin (30)}{x}$

Step 2

Factor each term.

Tap for more steps...

Step 2.1

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\frac{\sqrt{3}}{2}}{y} = \frac{\sin (30)}{x}$

Step 2.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{3}}{2} \cdot \frac{1}{y} = \frac{\sin (30)}{x}$

Step 2.3

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{1}{y}$ .

$\frac{\sqrt{3}}{2 y} = \frac{\sin (30)}{x}$

Step 2.4

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

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

Step 2.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.6

Multiply $\frac{1}{2}$ by $\frac{1}{x}$ .

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

Step 3

Find the LCD of the terms in the equation.

Tap for more steps...

Step 3.1

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

$2 y, 2 x$

Step 3.2

Since $2 y, 2 x$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $2, 2$ then find LCM for the variable part $y^{1}, x^{1}$ .

Step 3.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 3.4

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

Step 3.5

The LCM of $2, 2$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2$

Step 3.6

The factor for $y^{1}$ is $y$ itself.

$y^{1} = y$

$y$ occurs $1$ time.

Step 3.7

The factor for $x^{1}$ is $x$ itself.

$x^{1} = x$

$x$ occurs $1$ time.

Step 3.8

The LCM of $y^{1}, x^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$y \cdot x$

Step 3.9

Multiply $y$ by $x$ .

$y x$

Step 3.10

The LCM for $2 y, 2 x$ is the numeric part $2$ multiplied by the variable part.

$2 y x$

Step 4

Multiply each term in $\frac{\sqrt{3}}{2 y} = \frac{1}{2 x}$ by $2 y x$ to eliminate the fractions.

Tap for more steps...

Step 4.1

Multiply each term in $\frac{\sqrt{3}}{2 y} = \frac{1}{2 x}$ by $2 y x$ .

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

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Rewrite using the commutative property of multiplication.

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

Step 4.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.2.1

Factor $2$ out of $2 y$ .

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

Step 4.2.2.2

Cancel the common factor.

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

Step 4.2.2.3

Rewrite the expression.

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

Step 4.2.3

Cancel the common factor of $y$ .

Tap for more steps...

Step 4.2.3.1

Factor $y$ out of $y x$ .

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

Step 4.2.3.2

Cancel the common factor.

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

Step 4.2.3.3

Rewrite the expression.

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

Step 4.3

Simplify the right side.

Tap for more steps...

Step 4.3.1

Rewrite using the commutative property of multiplication.

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

Step 4.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.2.1

Factor $2$ out of $2 x$ .

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

Step 4.3.2.2

Cancel the common factor.

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

Step 4.3.2.3

Rewrite the expression.

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

Step 4.3.3

Cancel the common factor of $x$ .

Tap for more steps...

Step 4.3.3.1

Factor $x$ out of $y x$ .

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

Step 4.3.3.2

Cancel the common factor.

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

Step 4.3.3.3

Rewrite the expression.

$\sqrt{3} x = y$

Step 5

Rewrite the equation as $y = \sqrt{3} x$ .

$y = \sqrt{3} x$

Step 6

Interchange the variables.

$x = \sqrt{3} y$

Step 7

Solve for $y$ .

Tap for more steps...

Step 7.1

Rewrite the equation as $\sqrt{3} y = x$ .

$\sqrt{3} y = x$

Step 7.2

Divide each term in $\sqrt{3} y = x$ by $\sqrt{3}$ and simplify.

Tap for more steps...

Step 7.2.1

Divide each term in $\sqrt{3} y = x$ by $\sqrt{3}$ .

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

Step 7.2.2

Simplify the left side.

Tap for more steps...

Step 7.2.2.1

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

Tap for more steps...

Step 7.2.2.1.1

Cancel the common factor.

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

Step 7.2.2.1.2

Divide $y$ by $1$ .

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

Step 7.2.3

Simplify the right side.

Tap for more steps...

Step 7.2.3.1

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

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

Step 7.2.3.2

Combine and simplify the denominator.

Tap for more steps...

Step 7.2.3.2.1

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

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

Step 7.2.3.2.2

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

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

Step 7.2.3.2.3

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

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

Step 7.2.3.2.4

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

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

Step 7.2.3.2.5

Add $1$ and $1$ .

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

Step 7.2.3.2.6

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

Tap for more steps...

Step 7.2.3.2.6.1

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

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

Step 7.2.3.2.6.2

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

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

Step 7.2.3.2.6.3

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

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

Step 7.2.3.2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.2.3.2.6.4.1

Cancel the common factor.

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

Step 7.2.3.2.6.4.2

Rewrite the expression.

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

Step 7.2.3.2.6.5

Evaluate the exponent.

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

Step 8

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

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

Step 9

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

Tap for more steps...

Step 9.1

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

Step 9.2

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

Tap for more steps...

Step 9.2.1

Set up the composite result function.

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

Step 9.2.2

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

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

Step 9.2.3

Simplify the numerator.

Tap for more steps...

Step 9.2.3.1

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

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

Step 9.2.3.2

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

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

Step 9.2.3.3

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

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

Step 9.2.3.4

Add $1$ and $1$ .

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

Step 9.2.4

Simplify terms.

Tap for more steps...

Step 9.2.4.1

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

Tap for more steps...

Step 9.2.4.1.1

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

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

Step 9.2.4.1.2

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

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

Step 9.2.4.1.3

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

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

Step 9.2.4.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.2.4.1.4.1

Cancel the common factor.

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

Step 9.2.4.1.4.2

Rewrite the expression.

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

Step 9.2.4.1.5

Evaluate the exponent.

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

Step 9.2.4.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 9.2.4.2.1

Cancel the common factor.

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

Step 9.2.4.2.2

Divide $x$ by $1$ .

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

Step 9.3

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

Tap for more steps...

Step 9.3.1

Set up the composite result function.

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

Step 9.3.2

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

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

Step 9.3.3

Multiply $\sqrt{3} (\frac{x \sqrt{3}}{3})$ .

Tap for more steps...

Step 9.3.3.1

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

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

Step 9.3.3.2

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

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

Step 9.3.3.3

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

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

Step 9.3.3.4

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

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

Step 9.3.3.5

Add $1$ and $1$ .

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

Step 9.3.4

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

Tap for more steps...

Step 9.3.4.1

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

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

Step 9.3.4.2

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

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

Step 9.3.4.3

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

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

Step 9.3.4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.3.4.4.1

Cancel the common factor.

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

Step 9.3.4.4.2

Rewrite the expression.

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

Step 9.3.4.5

Evaluate the exponent.

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

Step 9.3.5

Cancel the common factor of $3$ .

Tap for more steps...

Step 9.3.5.1

Cancel the common factor.

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

Step 9.3.5.2

Divide $x$ by $1$ .

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

Step 9.4

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

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