Trigonometry Examples

$f (x) = 3 \cos (2 x + 3)$

Step 1

Write $f (x) = 3 \cos (2 x + 3)$ as an equation.

$y = 3 \cos (2 x + 3)$

Step 2

Interchange the variables.

$x = 3 \cos (2 y + 3)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $3 \cos (2 y + 3) = x$ .

$3 \cos (2 y + 3) = x$

Step 3.2

Divide each term in $3 \cos (2 y + 3) = x$ by $3$ and simplify.

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Step 3.2.1

Divide each term in $3 \cos (2 y + 3) = x$ by $3$ .

$\frac{3 \cos (2 y + 3)}{3} = \frac{x}{3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{3 \cos (2 y + 3)}{3} = \frac{x}{3}$

Step 3.2.2.1.2

Divide $\cos (2 y + 3)$ by $1$ .

$\cos (2 y + 3) = \frac{x}{3}$

Step 3.3

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

$2 y + 3 = \arccos (\frac{x}{3})$

Step 3.4

Subtract $3$ from both sides of the equation.

$2 y = \arccos (\frac{x}{3}) - 3$

Step 3.5

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

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Step 3.5.1

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

$\frac{2 y}{2} = \frac{\arccos (\frac{x}{3})}{2} + \frac{- 3}{2}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $2$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{\arccos (\frac{x}{3})}{2} + \frac{- 3}{2}$

Step 3.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{\arccos (\frac{x}{3})}{2} + \frac{- 3}{2}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Move the negative in front of the fraction.

$y = \frac{\arccos (\frac{x}{3})}{2} - \frac{3}{2}$

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \frac{\arccos (\frac{x}{3})}{2} - \frac{3}{2}$ is the inverse of $f (x) = 3 \cos (2 x + 3)$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

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

$f^{- 1} (3 \cos (2 x + 3)) = \frac{\arccos (\frac{3 \cos (2 x + 3)}{3})}{2} - \frac{3}{2}$

Step 5.2.3

Cancel the common factor of $3$ .

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Step 5.2.3.1

Cancel the common factor.

$f^{- 1} (3 \cos (2 x + 3)) = \frac{\arccos (\frac{3 \cos (2 x + 3)}{3})}{2} - \frac{3}{2}$

Step 5.2.3.2

Divide $\cos (2 x + 3)$ by $1$ .

$f^{- 1} (3 \cos (2 x + 3)) = \frac{\arccos (\cos (2 x + 3))}{2} - \frac{3}{2}$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

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

$f (\frac{\arccos (\frac{x}{3})}{2} - \frac{3}{2}) = 3 \cos (2 (\frac{\arccos (\frac{x}{3})}{2} - \frac{3}{2}) + 3)$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (\frac{\arccos (\frac{x}{3})}{2} - \frac{3}{2}) = 3 \cos (2 (\frac{\arccos (\frac{x}{3})}{2}) + 2 (- \frac{3}{2}) + 3)$

Step 5.3.3.2

Cancel the common factor of $2$ .

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Step 5.3.3.2.1

Cancel the common factor.

$f (\frac{\arccos (\frac{x}{3})}{2} - \frac{3}{2}) = 3 \cos (2 (\frac{\arccos (\frac{x}{3})}{2}) + 2 (- \frac{3}{2}) + 3)$

Step 5.3.3.2.2

Rewrite the expression.

$f (\frac{\arccos (\frac{x}{3})}{2} - \frac{3}{2}) = 3 \cos (\arccos (\frac{x}{3}) + 2 (- \frac{3}{2}) + 3)$

Step 5.3.3.3

Cancel the common factor of $2$ .

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Step 5.3.3.3.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$f (\frac{\arccos (\frac{x}{3})}{2} - \frac{3}{2}) = 3 \cos (\arccos (\frac{x}{3}) + 2 (\frac{- 3}{2}) + 3)$

Step 5.3.3.3.2

Cancel the common factor.

$f (\frac{\arccos (\frac{x}{3})}{2} - \frac{3}{2}) = 3 \cos (\arccos (\frac{x}{3}) + 2 (\frac{- 3}{2}) + 3)$

Step 5.3.3.3.3

Rewrite the expression.

$f (\frac{\arccos (\frac{x}{3})}{2} - \frac{3}{2}) = 3 \cos (\arccos (\frac{x}{3}) - 3 + 3)$

Step 5.3.4

Combine the opposite terms in $\arccos (\frac{x}{3}) - 3 + 3$ .

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Step 5.3.4.1

Add $- 3$ and $3$ .

$f (\frac{\arccos (\frac{x}{3})}{2} - \frac{3}{2}) = 3 \cos (\arccos (\frac{x}{3}) + 0)$

Step 5.3.4.2

Add $\arccos (\frac{x}{3})$ and $0$ .

$f (\frac{\arccos (\frac{x}{3})}{2} - \frac{3}{2}) = 3 \cos (\arccos (\frac{x}{3}))$

Step 5.3.5

The functions cosine and arccosine are inverses.

$f (\frac{\arccos (\frac{x}{3})}{2} - \frac{3}{2}) = 3 (\frac{x}{3})$

Step 5.3.6

Cancel the common factor of $3$ .

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Step 5.3.6.1

Cancel the common factor.

$f (\frac{\arccos (\frac{x}{3})}{2} - \frac{3}{2}) = 3 (\frac{x}{3})$

Step 5.3.6.2

Rewrite the expression.

$f (\frac{\arccos (\frac{x}{3})}{2} - \frac{3}{2}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\arccos (\frac{x}{3})}{2} - \frac{3}{2}$ is the inverse of $f (x) = 3 \cos (2 x + 3)$ .

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