Trigonometry Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 3$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

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

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide $y$ by $1$ .

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

Step 5

Interchange the variables.

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

Step 6

Solve for $y$ .

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Step 6.1

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

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

Step 6.2

Multiply both sides by $2$ .

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

Step 6.3

Simplify.

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Step 6.3.1

Simplify the left side.

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Step 6.3.1.1

Cancel the common factor of $2$ .

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Step 6.3.1.1.1

Cancel the common factor.

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

Step 6.3.1.1.2

Rewrite the expression.

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

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

Move $2$ to the left of $x$ .

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

Step 6.4

Solve for $y$ .

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Step 6.4.1

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

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

Step 6.4.2

Multiply both sides of the equation by $- 3$ .

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

Step 6.4.3

Simplify the left side.

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Step 6.4.3.1

Simplify $- 3 (- \frac{y}{3})$ .

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Step 6.4.3.1.1

Cancel the common factor of $3$ .

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Step 6.4.3.1.1.1

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

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

Step 6.4.3.1.1.2

Factor $3$ out of $- 3$ .

$3 (- 1) \frac{- y}{3} = - 3 \cos (2 x)$

Step 6.4.3.1.1.3

Cancel the common factor.

$3 \cdot - 1 \frac{- y}{3} = - 3 \cos (2 x)$

Step 6.4.3.1.1.4

Rewrite the expression.

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

Step 6.4.3.1.2

Multiply.

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Step 6.4.3.1.2.1

Multiply $- 1$ by $- 1$ .

$1 y = - 3 \cos (2 x)$

Step 6.4.3.1.2.2

Multiply $y$ by $1$ .

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

Step 7

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

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

Step 8

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

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Step 8.1

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

Step 8.2

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

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Step 8.2.1

Set up the composite result function.

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

Step 8.2.2

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

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

Step 8.2.3

Cancel the common factor of $2$ .

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Step 8.2.3.1

Cancel the common factor.

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

Step 8.2.3.2

Rewrite the expression.

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

Step 8.2.4

The functions cosine and arccosine are inverses.

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

Step 8.2.5

Cancel the common factor of $3$ .

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Step 8.2.5.1

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

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

Step 8.2.5.2

Factor $3$ out of $- 3$ .

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

Step 8.2.5.3

Cancel the common factor.

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

Step 8.2.5.4

Rewrite the expression.

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

Step 8.2.6

Multiply.

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Step 8.2.6.1

Multiply $- 1$ by $- 1$ .

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

Step 8.2.6.2

Multiply $x$ by $1$ .

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

Step 8.3

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

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Step 8.3.1

Set up the composite result function.

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

Step 8.3.2

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

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

Step 8.3.3

Cancel the common factor of $- 3$ and $3$ .

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Step 8.3.3.1

Factor $3$ out of $- 3 \cos (2 x)$ .

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

Step 8.3.3.2

Cancel the common factors.

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Step 8.3.3.2.1

Factor $3$ out of $3$ .

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

Step 8.3.3.2.2

Cancel the common factor.

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

Step 8.3.3.2.3

Rewrite the expression.

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

Step 8.3.3.2.4

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

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

Step 8.4

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

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