Trigonometry Examples

$4 \cos (3 x)$

Step 1

Interchange the variables.

$x = 4 \cos (3 y)$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $4 \cos (3 y) = x$ .

$4 \cos (3 y) = x$

Step 2.2

Divide each term in $4 \cos (3 y) = x$ by $4$ and simplify.

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Step 2.2.1

Divide each term in $4 \cos (3 y) = x$ by $4$ .

$\frac{4 \cos (3 y)}{4} = \frac{x}{4}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $4$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{4 \cos (3 y)}{4} = \frac{x}{4}$

Step 2.2.2.1.2

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

$\cos (3 y) = \frac{x}{4}$

Step 2.3

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

$3 y = \arccos (\frac{x}{4})$

Step 2.4

Divide each term in $3 y = \arccos (\frac{x}{4})$ by $3$ and simplify.

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Step 2.4.1

Divide each term in $3 y = \arccos (\frac{x}{4})$ by $3$ .

$\frac{3 y}{3} = \frac{\arccos (\frac{x}{4})}{3}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $3$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{\arccos (\frac{x}{4})}{3}$

Step 2.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{\arccos (\frac{x}{4})}{3}$

Step 3

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

$f^{- 1} (x) = \frac{\arccos (\frac{x}{4})}{3}$

Step 4

Verify if $f^{- 1} (x) = \frac{\arccos (\frac{x}{4})}{3}$ is the inverse of $f (x) = 4 \cos (3 x)$ .

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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 $f^{- 1} (4 \cos (3 x))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (4 \cos (3 x)) = \frac{\arccos (\frac{4 \cos (3 x)}{4})}{3}$

Step 4.2.3

Cancel the common factor of $4$ .

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Step 4.2.3.1

Cancel the common factor.

$f^{- 1} (4 \cos (3 x)) = \frac{\arccos (\frac{4 \cos (3 x)}{4})}{3}$

Step 4.2.3.2

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

$f^{- 1} (4 \cos (3 x)) = \frac{\arccos (\cos (3 x))}{3}$

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 $f (\frac{\arccos (\frac{x}{4})}{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\arccos (\frac{x}{4})}{3}) = 4 \cos (3 (\frac{\arccos (\frac{x}{4})}{3}))$

Step 4.3.3

Cancel the common factor of $3$ .

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Step 4.3.3.1

Cancel the common factor.

$f (\frac{\arccos (\frac{x}{4})}{3}) = 4 \cos (3 (\frac{\arccos (\frac{x}{4})}{3}))$

Step 4.3.3.2

Rewrite the expression.

$f (\frac{\arccos (\frac{x}{4})}{3}) = 4 \cos (\arccos (\frac{x}{4}))$

Step 4.3.4

The functions cosine and arccosine are inverses.

$f (\frac{\arccos (\frac{x}{4})}{3}) = 4 (\frac{x}{4})$

Step 4.3.5

Cancel the common factor of $4$ .

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Step 4.3.5.1

Cancel the common factor.

$f (\frac{\arccos (\frac{x}{4})}{3}) = 4 (\frac{x}{4})$

Step 4.3.5.2

Rewrite the expression.

$f (\frac{\arccos (\frac{x}{4})}{3}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\arccos (\frac{x}{4})}{3}$ is the inverse of $f (x) = 4 \cos (3 x)$ .

$f^{- 1} (x) = \frac{\arccos (\frac{x}{4})}{3}$