Trigonometry Examples

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{1}{1 - \sin (y)} = x$ .

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

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

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

$1 - \sin (y), 1$

Step 2.2.2

Remove parentheses.

$1 - \sin (y), 1$

Step 2.2.3

The LCM of one and any expression is the expression.

$1 - \sin (y)$

Step 2.3

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

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Step 2.3.1

Multiply each term in $\frac{1}{1 - \sin (y)} = x$ by $1 - \sin (y)$ .

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $1 - \sin (y)$ .

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Step 2.3.2.1.1

Cancel the common factor.

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

Step 2.3.2.1.2

Rewrite the expression.

$1 = x (1 - \sin (y))$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Apply the distributive property.

$1 = x \cdot 1 + x (- \sin (y))$

Step 2.3.3.2

Simplify the expression.

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Step 2.3.3.2.1

Multiply $x$ by $1$ .

$1 = x + x (- \sin (y))$

Step 2.3.3.2.2

Rewrite using the commutative property of multiplication.

$1 = x - x \sin (y)$

Step 2.4

Solve the equation.

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Step 2.4.1

Rewrite the equation as $x - x \sin (y) = 1$ .

$x - x \sin (y) = 1$

Step 2.4.2

Subtract $x$ from both sides of the equation.

$- x \sin (y) = 1 - x$

Step 2.4.3

Divide each term in $- x \sin (y) = 1 - x$ by $- x$ and simplify.

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Step 2.4.3.1

Divide each term in $- x \sin (y) = 1 - x$ by $- x$ .

$\frac{- x \sin (y)}{- x} = \frac{1}{- x} + \frac{- x}{- x}$

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Dividing two negative values results in a positive value.

$\frac{x \sin (y)}{x} = \frac{1}{- x} + \frac{- x}{- x}$

Step 2.4.3.2.2

Cancel the common factor of $x$ .

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Step 2.4.3.2.2.1

Cancel the common factor.

$\frac{x \sin (y)}{x} = \frac{1}{- x} + \frac{- x}{- x}$

Step 2.4.3.2.2.2

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

$\sin (y) = \frac{1}{- x} + \frac{- x}{- x}$

Step 2.4.3.3

Simplify the right side.

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Step 2.4.3.3.1

Simplify each term.

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Step 2.4.3.3.1.1

Cancel the common factor of $1$ and $- 1$ .

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Step 2.4.3.3.1.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$\sin (y) = \frac{- 1 (- 1)}{- x} + \frac{- x}{- x}$

Step 2.4.3.3.1.1.2

Move the negative in front of the fraction.

$\sin (y) = - \frac{1}{x} + \frac{- x}{- x}$

Step 2.4.3.3.1.2

Dividing two negative values results in a positive value.

$\sin (y) = - \frac{1}{x} + \frac{x}{x}$

Step 2.4.3.3.1.3

Cancel the common factor of $x$ .

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Step 2.4.3.3.1.3.1

Cancel the common factor.

$\sin (y) = - \frac{1}{x} + \frac{x}{x}$

Step 2.4.3.3.1.3.2

Rewrite the expression.

$\sin (y) = - \frac{1}{x} + 1$

Step 2.5

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

$y = \arcsin (- \frac{1}{x} + 1)$

Step 3

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

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

Step 4

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

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

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

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2.3.2

Multiply $1 - \sin (x)$ by $1$ .

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

Step 4.2.3.3

Apply the distributive property.

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

Step 4.2.3.4

Multiply $- 1$ by $1$ .

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

Step 4.2.3.5

Multiply $- - \sin (x)$ .

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Step 4.2.3.5.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2.3.5.2

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

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

Step 4.2.4

Combine the opposite terms in $- 1 + \sin (x) + 1$ .

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Step 4.2.4.1

Add $- 1$ and $1$ .

$f^{- 1} (\frac{1}{1 - \sin (x)}) = \arcsin (0 + \sin (x))$

Step 4.2.4.2

Add $0$ and $\sin (x)$ .

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

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

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

Step 4.3.3

Simplify the denominator.

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Step 4.3.3.1

The functions sine and arcsine are inverses.

$f (\arcsin (- \frac{1}{x} + 1)) = \frac{1}{1 - (- \frac{1}{x} + 1)}$

Step 4.3.3.2

Apply the distributive property.

$f (\arcsin (- \frac{1}{x} + 1)) = \frac{1}{1 + \frac{1}{x} - 1 \cdot 1}$

Step 4.3.3.3

Multiply $- - \frac{1}{x}$ .

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Step 4.3.3.3.1

Multiply $- 1$ by $- 1$ .

$f (\arcsin (- \frac{1}{x} + 1)) = \frac{1}{1 + 1 (\frac{1}{x}) - 1 \cdot 1}$

Step 4.3.3.3.2

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

$f (\arcsin (- \frac{1}{x} + 1)) = \frac{1}{1 + \frac{1}{x} - 1 \cdot 1}$

Step 4.3.3.4

Multiply $- 1$ by $1$ .

$f (\arcsin (- \frac{1}{x} + 1)) = \frac{1}{1 + \frac{1}{x} - 1}$

Step 4.3.3.5

Subtract $1$ from $1$ .

$f (\arcsin (- \frac{1}{x} + 1)) = \frac{1}{\frac{1}{x} + 0}$

Step 4.3.3.6

Add $\frac{1}{x}$ and $0$ .

$f (\arcsin (- \frac{1}{x} + 1)) = \frac{1}{\frac{1}{x}}$

Step 4.3.4

Multiply the numerator by the reciprocal of the denominator.

$f (\arcsin (- \frac{1}{x} + 1)) = 1 x$

Step 4.3.5

Multiply $x$ by $1$ .

$f (\arcsin (- \frac{1}{x} + 1)) = x$

Step 4.4

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

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