Trigonometry Examples

$- \tan (x + 7) - 4$

Step 1

Interchange the variables.

$x = - \tan (y + 7) - 4$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $- \tan (y + 7) - 4 = x$ .

$- \tan (y + 7) - 4 = x$

Step 2.2

Add $4$ to both sides of the equation.

$- \tan (y + 7) = x + 4$

Step 2.3

Divide each term in $- \tan (y + 7) = x + 4$ by $- 1$ and simplify.

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Step 2.3.1

Divide each term in $- \tan (y + 7) = x + 4$ by $- 1$ .

$\frac{- \tan (y + 7)}{- 1} = \frac{x}{- 1} + \frac{4}{- 1}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

$\frac{\tan (y + 7)}{1} = \frac{x}{- 1} + \frac{4}{- 1}$

Step 2.3.2.2

Divide $\tan (y + 7)$ by $1$ .

$\tan (y + 7) = \frac{x}{- 1} + \frac{4}{- 1}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Move the negative one from the denominator of $\frac{x}{- 1}$ .

$\tan (y + 7) = - 1 \cdot x + \frac{4}{- 1}$

Step 2.3.3.1.2

Rewrite $- 1 \cdot x$ as $- x$ .

$\tan (y + 7) = - x + \frac{4}{- 1}$

Step 2.3.3.1.3

Divide $4$ by $- 1$ .

$\tan (y + 7) = - x - 4$

Step 2.4

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

$y + 7 = \arctan (- x - 4)$

Step 2.5

Subtract $7$ from both sides of the equation.

$y = \arctan (- x - 4) - 7$

Step 3

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

$f^{- 1} (x) = \arctan (- x - 4) - 7$

Step 4

Verify if $f^{- 1} (x) = \arctan (- x - 4) - 7$ is the inverse of $f (x) = - \tan (x + 7) - 4$ .

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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 $f^{- 1} (- \tan (x + 7) - 4)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \tan (x + 7) - 4) = \arctan (- (- \tan (x + 7) - 4) - 4) - 7$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Apply the distributive property.

$f^{- 1} (- \tan (x + 7) - 4) = \arctan (\tan (x + 7) + 4 - 4) - 7$

Step 4.2.3.1.2

Multiply $- - \tan (x + 7)$ .

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Step 4.2.3.1.2.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (- \tan (x + 7) - 4) = \arctan (1 \tan (x + 7) + 4 - 4) - 7$

Step 4.2.3.1.2.2

Multiply $\tan (x + 7)$ by $1$ .

$f^{- 1} (- \tan (x + 7) - 4) = \arctan (\tan (x + 7) + 4 - 4) - 7$

Step 4.2.3.1.3

Multiply $- 1$ by $- 4$ .

$f^{- 1} (- \tan (x + 7) - 4) = \arctan (\tan (x + 7) + 4 - 4) - 7$

Step 4.2.3.2

Combine the opposite terms in $\tan (x + 7) + 4 - 4$ .

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Step 4.2.3.2.1

Subtract $4$ from $4$ .

$f^{- 1} (- \tan (x + 7) - 4) = \arctan (\tan (x + 7) + 0) - 7$

Step 4.2.3.2.2

Add $\tan (x + 7)$ and $0$ .

$f^{- 1} (- \tan (x + 7) - 4) = \arctan (\tan (x + 7)) - 7$

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 $f (\arctan (- x - 4) - 7)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\arctan (- x - 4) - 7) = - \tan ((\arctan (- x - 4) - 7) + 7) - 4$

Step 4.3.3

Combine the opposite terms in $- \tan ((\arctan (- x - 4) - 7) + 7) - 4$ .

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Step 4.3.3.1

Add $- 7$ and $7$ .

$f (\arctan (- x - 4) - 7) = - \tan (\arctan (- x - 4) + 0) - 4$

Step 4.3.3.2

Add $\arctan (- x - 4)$ and $0$ .

$f (\arctan (- x - 4) - 7) = - \tan (\arctan (- x - 4)) - 4$

Step 4.3.4

Simplify each term.

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Step 4.3.4.1

The functions tangent and arctangent are inverses.

$f (\arctan (- x - 4) - 7) = - (- x - 4) - 4$

Step 4.3.4.2

Apply the distributive property.

$f (\arctan (- x - 4) - 7) = x + 4 - 4$

Step 4.3.4.3

Multiply $- - x$ .

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Step 4.3.4.3.1

Multiply $- 1$ by $- 1$ .

$f (\arctan (- x - 4) - 7) = 1 x + 4 - 4$

Step 4.3.4.3.2

Multiply $x$ by $1$ .

$f (\arctan (- x - 4) - 7) = x + 4 - 4$

Step 4.3.4.4

Multiply $- 1$ by $- 4$ .

$f (\arctan (- x - 4) - 7) = x + 4 - 4$

Step 4.3.5

Combine the opposite terms in $x + 4 - 4$ .

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Step 4.3.5.1

Subtract $4$ from $4$ .

$f (\arctan (- x - 4) - 7) = x + 0$

Step 4.3.5.2

Add $x$ and $0$ .

$f (\arctan (- x - 4) - 7) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \arctan (- x - 4) - 7$ is the inverse of $f (x) = - \tan (x + 7) - 4$ .

$f^{- 1} (x) = \arctan (- x - 4) - 7$