Trigonometry Examples

$- 2 \cos (3 x - \frac{π}{4})$

Step 1

Interchange the variables.

$x = - 2 \cos (3 y - \frac{π}{4})$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $- 2 \cos (3 y - \frac{π}{4}) = x$ .

$- 2 \cos (3 y - \frac{π}{4}) = x$

Step 2.2

Divide each term in $- 2 \cos (3 y - \frac{π}{4}) = x$ by $- 2$ and simplify.

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Step 2.2.1

Divide each term in $- 2 \cos (3 y - \frac{π}{4}) = x$ by $- 2$ .

$\frac{- 2 \cos (3 y - \frac{π}{4})}{- 2} = \frac{x}{- 2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 2 \cos (3 y - \frac{π}{4})}{- 2} = \frac{x}{- 2}$

Step 2.2.2.1.2

Divide $\cos (3 y - \frac{π}{4})$ by $1$ .

$\cos (3 y - \frac{π}{4}) = \frac{x}{- 2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

$\cos (3 y - \frac{π}{4}) = - \frac{x}{2}$

Step 2.3

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

$3 y - \frac{π}{4} = \arccos (- \frac{x}{2})$

Step 2.4

Add $\frac{π}{4}$ to both sides of the equation.

$3 y = \arccos (- \frac{x}{2}) + \frac{π}{4}$

Step 2.5

Divide each term in $3 y = \arccos (- \frac{x}{2}) + \frac{π}{4}$ by $3$ and simplify.

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Step 2.5.1

Divide each term in $3 y = \arccos (- \frac{x}{2}) + \frac{π}{4}$ by $3$ .

$\frac{3 y}{3} = \frac{\arccos (- \frac{x}{2})}{3} + \frac{\frac{π}{4}}{3}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $3$ .

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Step 2.5.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{\arccos (- \frac{x}{2})}{3} + \frac{\frac{π}{4}}{3}$

Step 2.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{\arccos (- \frac{x}{2})}{3} + \frac{\frac{π}{4}}{3}$

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Simplify each term.

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Step 2.5.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{4} \cdot \frac{1}{3}$

Step 2.5.3.1.2

Multiply $\frac{π}{4} \cdot \frac{1}{3}$ .

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Step 2.5.3.1.2.1

Multiply $\frac{π}{4}$ by $\frac{1}{3}$ .

$y = \frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{4 \cdot 3}$

Step 2.5.3.1.2.2

Multiply $4$ by $3$ .

$y = \frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}$

Step 3

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

$f^{- 1} (x) = \frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}$

Step 4

Verify if $f^{- 1} (x) = \frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}$ is the inverse of $f (x) = - 2 \cos (3 x - \frac{π}{4})$ .

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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 $f^{- 1} (- 2 \cos (3 x - \frac{π}{4}))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- 2 \cos (3 x - \frac{π}{4})) = \frac{\arccos (- \frac{- 2 \cos (3 x - \frac{π}{4})}{2})}{3} + \frac{π}{12}$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Cancel the common factor of $- 2$ and $2$ .

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Step 4.2.3.1.1

Factor $2$ out of $- 2 \cos (3 x - \frac{π}{4})$ .

$f^{- 1} (- 2 \cos (3 x - \frac{π}{4})) = \frac{\arccos (- \frac{2 (- \cos (3 x - \frac{π}{4}))}{2})}{3} + \frac{π}{12}$

Step 4.2.3.1.2

Cancel the common factors.

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Step 4.2.3.1.2.1

Factor $2$ out of $2$ .

$f^{- 1} (- 2 \cos (3 x - \frac{π}{4})) = \frac{\arccos (- \frac{2 (- \cos (3 x - \frac{π}{4}))}{2 (1)})}{3} + \frac{π}{12}$

Step 4.2.3.1.2.2

Cancel the common factor.

$f^{- 1} (- 2 \cos (3 x - \frac{π}{4})) = \frac{\arccos (- \frac{2 (- \cos (3 x - \frac{π}{4}))}{2 \cdot 1})}{3} + \frac{π}{12}$

Step 4.2.3.1.2.3

Rewrite the expression.

$f^{- 1} (- 2 \cos (3 x - \frac{π}{4})) = \frac{\arccos (- \frac{- \cos (3 x - \frac{π}{4})}{1})}{3} + \frac{π}{12}$

Step 4.2.3.1.2.4

Divide $- \cos (3 x - \frac{π}{4})$ by $1$ .

$f^{- 1} (- 2 \cos (3 x - \frac{π}{4})) = \frac{\arccos (\cos (3 x - \frac{π}{4}))}{3} + \frac{π}{12}$

Step 4.2.3.2

Multiply $- - \cos (3 x - \frac{π}{4})$ .

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Step 4.2.3.2.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (- 2 \cos (3 x - \frac{π}{4})) = \frac{\arccos (1 \cos (3 x - \frac{π}{4}))}{3} + \frac{π}{12}$

Step 4.2.3.2.2

Multiply $\cos (3 x - \frac{π}{4})$ by $1$ .

$f^{- 1} (- 2 \cos (3 x - \frac{π}{4})) = \frac{\arccos (\cos (3 x - \frac{π}{4}))}{3} + \frac{π}{12}$

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 $f (\frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}) = - 2 \cos (3 (\frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}) - \frac{π}{4})$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

$f (\frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}) = - 2 \cos (3 (\frac{\arccos (- \frac{x}{2})}{3}) + 3 (\frac{π}{12}) - \frac{π}{4})$

Step 4.3.3.2

Cancel the common factor of $3$ .

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Step 4.3.3.2.1

Cancel the common factor.

$f (\frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}) = - 2 \cos (3 (\frac{\arccos (- \frac{x}{2})}{3}) + 3 (\frac{π}{12}) - \frac{π}{4})$

Step 4.3.3.2.2

Rewrite the expression.

$f (\frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}) = - 2 \cos (\arccos (- \frac{x}{2}) + 3 (\frac{π}{12}) - \frac{π}{4})$

Step 4.3.3.3

Cancel the common factor of $3$ .

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Step 4.3.3.3.1

Factor $3$ out of $12$ .

$f (\frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}) = - 2 \cos (\arccos (- \frac{x}{2}) + 3 (\frac{π}{3 (4)}) - \frac{π}{4})$

Step 4.3.3.3.2

Cancel the common factor.

$f (\frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}) = - 2 \cos (\arccos (- \frac{x}{2}) + 3 (\frac{π}{3 \cdot 4}) - \frac{π}{4})$

Step 4.3.3.3.3

Rewrite the expression.

$f (\frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}) = - 2 \cos (\arccos (- \frac{x}{2}) + \frac{π}{4} - \frac{π}{4})$

Step 4.3.4

Simplify by adding terms.

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Step 4.3.4.1

Combine the opposite terms in $\arccos (- \frac{x}{2}) + \frac{π}{4} - \frac{π}{4}$ .

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Step 4.3.4.1.1

Combine the numerators over the common denominator.

$f (\frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}) = - 2 \cos (\arccos (- \frac{x}{2}) + \frac{π - π}{4})$

Step 4.3.4.1.2

Subtract $π$ from $π$ .

$f (\frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}) = - 2 \cos (\arccos (- \frac{x}{2}) + \frac{0}{4})$

Step 4.3.4.2

Simplify the expression.

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Step 4.3.4.2.1

Divide $0$ by $4$ .

$f (\frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}) = - 2 \cos (\arccos (- \frac{x}{2}) + 0)$

Step 4.3.4.2.2

Add $\arccos (- \frac{x}{2})$ and $0$ .

$f (\frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}) = - 2 \cos (\arccos (- \frac{x}{2}))$

Step 4.3.5

The functions cosine and arccosine are inverses.

$f (\frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}) = - 2 (- \frac{x}{2})$

Step 4.3.6

Cancel the common factor of $2$ .

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Step 4.3.6.1

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

$f (\frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}) = - 2 \frac{- x}{2}$

Step 4.3.6.2

Factor $2$ out of $- 2$ .

$f (\frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}) = 2 (- 1) (\frac{- x}{2})$

Step 4.3.6.3

Cancel the common factor.

$f (\frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}) = 2 \cdot (- 1 \frac{- x}{2})$

Step 4.3.6.4

Rewrite the expression.

$f (\frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}) = - 1 (- x)$

Step 4.3.7

Multiply.

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Step 4.3.7.1

Multiply $- 1$ by $- 1$ .

$f (\frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}) = 1 x$

Step 4.3.7.2

Multiply $x$ by $1$ .

$f (\frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}$ is the inverse of $f (x) = - 2 \cos (3 x - \frac{π}{4})$ .

$f^{- 1} (x) = \frac{\arccos (- \frac{x}{2})}{3} + \frac{π}{12}$