Trigonometry Examples

$4 \cos (\frac{x}{2} - \frac{π}{3})$

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

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.

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

Step 2.2.2.1.2

Divide $\cos (\frac{y}{2} - \frac{π}{3})$ by $1$ .

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

Step 2.3

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

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

Step 2.4

Add $\frac{π}{3}$ to both sides of the equation.

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

Step 2.5

Multiply both sides of the equation by $2$ .

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

Step 2.6

Simplify both sides of the equation.

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Step 2.6.1

Simplify the left side.

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Step 2.6.1.1

Cancel the common factor of $2$ .

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Step 2.6.1.1.1

Cancel the common factor.

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

Step 2.6.1.1.2

Rewrite the expression.

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

Step 2.6.2

Simplify the right side.

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Step 2.6.2.1

Simplify $2 (\arccos (\frac{x}{4}) + \frac{π}{3})$ .

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Step 2.6.2.1.1

Apply the distributive property.

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

Step 2.6.2.1.2

Combine $2$ and $\frac{π}{3}$ .

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

Step 2.7

Add $\frac{π}{3}$ to both sides of the equation.

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

Step 2.8

Multiply both sides of the equation by $2$ .

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

Step 2.9

Simplify both sides of the equation.

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Step 2.9.1

Simplify the left side.

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Step 2.9.1.1

Cancel the common factor of $2$ .

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Step 2.9.1.1.1

Cancel the common factor.

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

Step 2.9.1.1.2

Rewrite the expression.

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

Step 2.9.2

Simplify the right side.

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Step 2.9.2.1

Simplify $2 (\arccos (\frac{x}{4}) + \frac{π}{3})$ .

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Step 2.9.2.1.1

Apply the distributive property.

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

Step 2.9.2.1.2

Combine $2$ and $\frac{π}{3}$ .

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

Step 3

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

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

Step 4

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

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

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

Step 4.2.3

Cancel the common factor of $4$ .

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Step 4.2.3.1

Cancel the common factor.

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

Step 4.2.3.2

Divide $\cos (\frac{x}{2} - \frac{π}{3})$ by $1$ .

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

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

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

Step 4.3.3

Cancel the common factor of $2 \arccos (\frac{x}{4}) + \frac{2 π}{3}$ and $2$ .

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Step 4.3.3.1

Factor $2$ out of $2 \arccos (\frac{x}{4})$ .

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

Step 4.3.3.2

Factor $2$ out of $\frac{2 π}{3}$ .

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

Step 4.3.3.3

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

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

Step 4.3.3.4

Cancel the common factors.

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Step 4.3.3.4.1

Factor $2$ out of $2$ .

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

Step 4.3.3.4.2

Cancel the common factor.

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

Step 4.3.3.4.3

Rewrite the expression.

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

Step 4.3.3.4.4

Divide $\arccos (\frac{x}{4}) + \frac{π}{3}$ by $1$ .

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

Step 4.3.4

Combine the opposite terms in $\arccos (\frac{x}{4}) + \frac{π}{3} - \frac{π}{3}$ .

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Step 4.3.4.1

Combine the numerators over the common denominator.

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

Step 4.3.4.2

Subtract $π$ from $π$ .

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

Step 4.3.5

Simplify the expression.

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Step 4.3.5.1

Divide $0$ by $3$ .

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

Step 4.3.5.2

Add $\arccos (\frac{x}{4})$ and $0$ .

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

Step 4.3.6

The functions cosine and arccosine are inverses.

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

Step 4.3.7

Cancel the common factor of $4$ .

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Step 4.3.7.1

Cancel the common factor.

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

Step 4.3.7.2

Rewrite the expression.

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

Step 4.4

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

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