Trigonometry Examples

$10 \cos (2 x) + 10$

Step 1

Interchange the variables.

$x = 10 \cos (2 y) + 10$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $10 \cos (2 y) + 10 = x$ .

$10 \cos (2 y) + 10 = x$

Step 2.2

Subtract $10$ from both sides of the equation.

$10 \cos (2 y) = x - 10$

Step 2.3

Divide each term in $10 \cos (2 y) = x - 10$ by $10$ and simplify.

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Step 2.3.1

Divide each term in $10 \cos (2 y) = x - 10$ by $10$ .

$\frac{10 \cos (2 y)}{10} = \frac{x}{10} + \frac{- 10}{10}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $10$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{10 \cos (2 y)}{10} = \frac{x}{10} + \frac{- 10}{10}$

Step 2.3.2.1.2

Divide $\cos (2 y)$ by $1$ .

$\cos (2 y) = \frac{x}{10} + \frac{- 10}{10}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $- 10$ by $10$ .

$\cos (2 y) = \frac{x}{10} - 1$

Step 2.4

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

$2 y = \arccos (\frac{x}{10} - 1)$

Step 2.5

Divide each term in $2 y = \arccos (\frac{x}{10} - 1)$ by $2$ and simplify.

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Step 2.5.1

Divide each term in $2 y = \arccos (\frac{x}{10} - 1)$ by $2$ .

$\frac{2 y}{2} = \frac{\arccos (\frac{x}{10} - 1)}{2}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $2$ .

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Step 2.5.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{\arccos (\frac{x}{10} - 1)}{2}$

Step 2.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{\arccos (\frac{x}{10} - 1)}{2}$

Step 3

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

$f^{- 1} (x) = \frac{\arccos (\frac{x}{10} - 1)}{2}$

Step 4

Verify if $f^{- 1} (x) = \frac{\arccos (\frac{x}{10} - 1)}{2}$ is the inverse of $f (x) = 10 \cos (2 x) + 10$ .

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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 $f^{- 1} (10 \cos (2 x) + 10)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (10 \cos (2 x) + 10) = \frac{\arccos (\frac{10 \cos (2 x) + 10}{10} - 1)}{2}$

Step 4.2.3

Cancel the common factor of $10 \cos (2 x) + 10$ and $10$ .

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Step 4.2.3.1

Factor $10$ out of $10 \cos (2 x)$ .

$f^{- 1} (10 \cos (2 x) + 10) = \frac{\arccos (\frac{10 (\cos (2 x)) + 10}{10} - 1)}{2}$

Step 4.2.3.2

Factor $10$ out of $10$ .

$f^{- 1} (10 \cos (2 x) + 10) = \frac{\arccos (\frac{10 (\cos (2 x)) + 10 \cdot 1}{10} - 1)}{2}$

Step 4.2.3.3

Factor $10$ out of $10 (\cos (2 x)) + 10 (1)$ .

$f^{- 1} (10 \cos (2 x) + 10) = \frac{\arccos (\frac{10 (\cos (2 x) + 1)}{10} - 1)}{2}$

Step 4.2.3.4

Cancel the common factors.

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Step 4.2.3.4.1

Factor $10$ out of $10$ .

$f^{- 1} (10 \cos (2 x) + 10) = \frac{\arccos (\frac{10 (\cos (2 x) + 1)}{10 (1)} - 1)}{2}$

Step 4.2.3.4.2

Cancel the common factor.

$f^{- 1} (10 \cos (2 x) + 10) = \frac{\arccos (\frac{10 (\cos (2 x) + 1)}{10 \cdot 1} - 1)}{2}$

Step 4.2.3.4.3

Rewrite the expression.

$f^{- 1} (10 \cos (2 x) + 10) = \frac{\arccos (\frac{\cos (2 x) + 1}{1} - 1)}{2}$

Step 4.2.3.4.4

Divide $\cos (2 x) + 1$ by $1$ .

$f^{- 1} (10 \cos (2 x) + 10) = \frac{\arccos (\cos (2 x) + 1 - 1)}{2}$

Step 4.2.4

Combine the opposite terms in $\cos (2 x) + 1 - 1$ .

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Step 4.2.4.1

Subtract $1$ from $1$ .

$f^{- 1} (10 \cos (2 x) + 10) = \frac{\arccos (\cos (2 x) + 0)}{2}$

Step 4.2.4.2

Add $\cos (2 x)$ and $0$ .

$f^{- 1} (10 \cos (2 x) + 10) = \frac{\arccos (\cos (2 x))}{2}$

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 $f (\frac{\arccos (\frac{x}{10} - 1)}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\arccos (\frac{x}{10} - 1)}{2}) = 10 \cos (2 (\frac{\arccos (\frac{x}{10} - 1)}{2})) + 10$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Simplify the numerator.

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Step 4.3.3.1.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$f (\frac{\arccos (\frac{x}{10} - 1)}{2}) = 10 \cos (2 (\frac{\arccos (\frac{x}{10} - 1 \cdot \frac{10}{10})}{2})) + 10$

Step 4.3.3.1.2

Combine $- 1$ and $\frac{10}{10}$ .

$f (\frac{\arccos (\frac{x}{10} - 1)}{2}) = 10 \cos (2 (\frac{\arccos (\frac{x}{10} + \frac{- 1 \cdot 10}{10})}{2})) + 10$

Step 4.3.3.1.3

Combine the numerators over the common denominator.

$f (\frac{\arccos (\frac{x}{10} - 1)}{2}) = 10 \cos (2 (\frac{\arccos (\frac{x - 1 \cdot 10}{10})}{2})) + 10$

Step 4.3.3.1.4

Multiply $- 1$ by $10$ .

$f (\frac{\arccos (\frac{x}{10} - 1)}{2}) = 10 \cos (2 (\frac{\arccos (\frac{x - 10}{10})}{2})) + 10$

Step 4.3.3.2

Cancel the common factor of $2$ .

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Step 4.3.3.2.1

Cancel the common factor.

$f (\frac{\arccos (\frac{x}{10} - 1)}{2}) = 10 \cos (2 (\frac{\arccos (\frac{x - 10}{10})}{2})) + 10$

Step 4.3.3.2.2

Rewrite the expression.

$f (\frac{\arccos (\frac{x}{10} - 1)}{2}) = 10 \cos (\arccos (\frac{x - 10}{10})) + 10$

Step 4.3.3.3

The functions cosine and arccosine are inverses.

$f (\frac{\arccos (\frac{x}{10} - 1)}{2}) = 10 (\frac{x - 10}{10}) + 10$

Step 4.3.3.4

Cancel the common factor of $10$ .

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Step 4.3.3.4.1

Cancel the common factor.

$f (\frac{\arccos (\frac{x}{10} - 1)}{2}) = 10 (\frac{x - 10}{10}) + 10$

Step 4.3.3.4.2

Rewrite the expression.

$f (\frac{\arccos (\frac{x}{10} - 1)}{2}) = x - 10 + 10$

Step 4.3.4

Combine the opposite terms in $x - 10 + 10$ .

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Step 4.3.4.1

Add $- 10$ and $10$ .

$f (\frac{\arccos (\frac{x}{10} - 1)}{2}) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (\frac{\arccos (\frac{x}{10} - 1)}{2}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\arccos (\frac{x}{10} - 1)}{2}$ is the inverse of $f (x) = 10 \cos (2 x) + 10$ .

$f^{- 1} (x) = \frac{\arccos (\frac{x}{10} - 1)}{2}$