Trigonometry Examples

$\cos (2) x - \frac{π}{3}$

Step 1

Interchange the variables.

$x = \cos (2) y - \frac{π}{3}$

Step 2

Solve for $y$ .

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Step 2.1

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

$\cos (2) y - \frac{π}{3} = x$

Step 2.2

Simplify the left side.

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Step 2.2.1

Evaluate $\cos (2)$ .

$- 0.41614683 y - \frac{π}{3} = x$

Step 2.3

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

$- 0.41614683 y = x + \frac{π}{3}$

Step 2.4

Divide each term in $- 0.41614683 y = x + \frac{π}{3}$ by $- 0.41614683$ and simplify.

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Step 2.4.1

Divide each term in $- 0.41614683 y = x + \frac{π}{3}$ by $- 0.41614683$ .

$\frac{- 0.41614683 y}{- 0.41614683} = \frac{x}{- 0.41614683} + \frac{\frac{π}{3}}{- 0.41614683}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $- 0.41614683$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{- 0.41614683 y}{- 0.41614683} = \frac{x}{- 0.41614683} + \frac{\frac{π}{3}}{- 0.41614683}$

Step 2.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{- 0.41614683} + \frac{\frac{π}{3}}{- 0.41614683}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

Move the negative in front of the fraction.

$y = - \frac{x}{0.41614683} + \frac{\frac{π}{3}}{- 0.41614683}$

Step 2.4.3.1.2

Multiply by $1$ .

$y = - \frac{1 x}{0.41614683} + \frac{\frac{π}{3}}{- 0.41614683}$

Step 2.4.3.1.3

Factor $0.41614683$ out of $0.41614683$ .

$y = - \frac{1 x}{0.41614683 (1)} + \frac{\frac{π}{3}}{- 0.41614683}$

Step 2.4.3.1.4

Separate fractions.

$y = - (\frac{1}{0.41614683} \cdot \frac{x}{1}) + \frac{\frac{π}{3}}{- 0.41614683}$

Step 2.4.3.1.5

Divide $1$ by $0.41614683$ .

$y = - (2.40299796 \frac{x}{1}) + \frac{\frac{π}{3}}{- 0.41614683}$

Step 2.4.3.1.6

Divide $x$ by $1$ .

$y = - (2.40299796 x) + \frac{\frac{π}{3}}{- 0.41614683}$

Step 2.4.3.1.7

Multiply $2.40299796$ by $- 1$ .

$y = - 2.40299796 x + \frac{\frac{π}{3}}{- 0.41614683}$

Step 2.4.3.1.8

Multiply the numerator by the reciprocal of the denominator.

$y = - 2.40299796 x + \frac{π}{3} \cdot \frac{1}{- 0.41614683}$

Step 2.4.3.1.9

Divide $1$ by $- 0.41614683$ .

$y = - 2.40299796 x + \frac{π}{3} \cdot - 2.40299796$

Step 2.4.3.1.10

Multiply $\frac{π}{3} \cdot - 2.40299796$ .

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Step 2.4.3.1.10.1

Combine $\frac{π}{3}$ and $- 2.40299796$ .

$y = - 2.40299796 x + \frac{π \cdot - 2.40299796}{3}$

Step 2.4.3.1.10.2

Multiply $π$ by $- 2.40299796$ .

$y = - 2.40299796 x + \frac{- 7.54924074}{3}$

Step 2.4.3.1.11

Divide $- 7.54924074$ by $3$ .

$y = - 2.40299796 x - 2.51641358$

Step 3

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

$f^{- 1} (x) = - 2.40299796 x - 2.51641358$

Step 4

Verify if $f^{- 1} (x) = - 2.40299796 x - 2.51641358$ is the inverse of $f (x) = \cos (2) x - \frac{π}{3}$ .

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

$f^{- 1} (\cos (2) x - \frac{π}{3}) = - 2.40299796 (\cos (2) x - \frac{π}{3}) - 2.51641358$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Evaluate $\cos (2)$ .

$f^{- 1} (\cos (2) x - \frac{π}{3}) = - 2.40299796 (- 0.41614683 x - \frac{π}{3}) - 2.51641358$

Step 4.2.3.2

Apply the distributive property.

$f^{- 1} (\cos (2) x - \frac{π}{3}) = - 2.40299796 (- 0.41614683 x) - 2.40299796 (- \frac{π}{3}) - 2.51641358$

Step 4.2.3.3

Multiply $- 2.40299796 (- 0.41614683 x)$ .

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Step 4.2.3.3.1

Multiply $- 0.41614683$ by $- 2.40299796$ .

$f^{- 1} (\cos (2) x - \frac{π}{3}) = 1 x - 2.40299796 (- \frac{π}{3}) - 2.51641358$

Step 4.2.3.3.2

Multiply $x$ by $1$ .

$f^{- 1} (\cos (2) x - \frac{π}{3}) = x - 2.40299796 (- \frac{π}{3}) - 2.51641358$

Step 4.2.3.4

Multiply $- 2.40299796 (- \frac{π}{3})$ .

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Step 4.2.3.4.1

Multiply $- 1$ by $- 2.40299796$ .

$f^{- 1} (\cos (2) x - \frac{π}{3}) = x + 2.40299796 (\frac{π}{3}) - 2.51641358$

Step 4.2.3.4.2

Combine $2.40299796$ and $\frac{π}{3}$ .

$f^{- 1} (\cos (2) x - \frac{π}{3}) = x + \frac{2.40299796 π}{3} - 2.51641358$

Step 4.2.3.4.3

Multiply $2.40299796$ by $π$ .

$f^{- 1} (\cos (2) x - \frac{π}{3}) = x + \frac{7.54924074}{3} - 2.51641358$

Step 4.2.3.5

Divide $7.54924074$ by $3$ .

$f^{- 1} (\cos (2) x - \frac{π}{3}) = x + 2.51641358 - 2.51641358$

Step 4.2.4

Combine the opposite terms in $x + 2.51641358 - 2.51641358$ .

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Step 4.2.4.1

Subtract $2.51641358$ from $2.51641358$ .

$f^{- 1} (\cos (2) x - \frac{π}{3}) = x + 0$

Step 4.2.4.2

Add $x$ and $0$ .

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

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 $f (- 2.40299796 x - 2.51641358)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- 2.40299796 x - 2.51641358) = \cos (2) (- 2.40299796 x - 2.51641358) - \frac{π}{3}$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Evaluate $\cos (2)$ .

$f (- 2.40299796 x - 2.51641358) = - 0.41614683 (- 2.40299796 x - 2.51641358) - \frac{π}{3}$

Step 4.3.3.2

Apply the distributive property.

$f (- 2.40299796 x - 2.51641358) = - 0.41614683 (- 2.40299796 x) - 0.41614683 \cdot - 2.51641358 - \frac{π}{3}$

Step 4.3.3.3

Multiply $- 0.41614683 (- 2.40299796 x)$ .

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Step 4.3.3.3.1

Multiply $- 2.40299796$ by $- 0.41614683$ .

$f (- 2.40299796 x - 2.51641358) = 1 x - 0.41614683 \cdot - 2.51641358 - \frac{π}{3}$

Step 4.3.3.3.2

Multiply $x$ by $1$ .

$f (- 2.40299796 x - 2.51641358) = x - 0.41614683 \cdot - 2.51641358 - \frac{π}{3}$

Step 4.3.3.4

Multiply $- 0.41614683$ by $- 2.51641358$ .

$f (- 2.40299796 x - 2.51641358) = x + 1.04719755 - \frac{π}{3}$

Step 4.3.4

Subtract $\frac{π}{3}$ from $1.04719755$ .

$f (- 2.40299796 x - 2.51641358) = x 0$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - 2.40299796 x - 2.51641358$ is the inverse of $f (x) = \cos (2) x - \frac{π}{3}$ .

$f^{- 1} (x) = - 2.40299796 x - 2.51641358$