Trigonometry Examples

$\frac{\tan (x) \cdot 5}{8}$

Step 1

Interchange the variables.

$x = \frac{\tan (y) \cdot 5}{8}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{\tan (y) \cdot 5}{8} = x$ .

$\frac{\tan (y) \cdot 5}{8} = x$

Step 2.2

Multiply both sides of the equation by $\frac{8}{5}$ .

$\frac{8}{5} \cdot \frac{\tan (y) \cdot 5}{8} = \frac{8}{5} x$

Step 2.3

Simplify both sides of the equation.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Simplify $\frac{8}{5} \cdot \frac{\tan (y) \cdot 5}{8}$ .

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Step 2.3.1.1.1

Cancel the common factor of $8$ .

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Step 2.3.1.1.1.1

Cancel the common factor.

$\frac{8}{5} \cdot \frac{\tan (y) \cdot 5}{8} = \frac{8}{5} x$

Step 2.3.1.1.1.2

Rewrite the expression.

$\frac{1}{5} (\tan (y) \cdot 5) = \frac{8}{5} x$

Step 2.3.1.1.2

Cancel the common factor of $5$ .

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Step 2.3.1.1.2.1

Factor $5$ out of $\tan (y) \cdot 5$ .

$\frac{1}{5} (5 \cdot \tan (y)) = \frac{8}{5} x$

Step 2.3.1.1.2.2

Cancel the common factor.

$\frac{1}{5} (5 \cdot \tan (y)) = \frac{8}{5} x$

Step 2.3.1.1.2.3

Rewrite the expression.

$\tan (y) = \frac{8}{5} x$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Combine $\frac{8}{5}$ and $x$ .

$\tan (y) = \frac{8 x}{5}$

Step 2.4

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

$y = \arctan (\frac{8 x}{5})$

Step 3

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

$f^{- 1} (x) = \arctan (\frac{8 x}{5})$

Step 4

Verify if $f^{- 1} (x) = \arctan (\frac{8 x}{5})$ is the inverse of $f (x) = \frac{\tan (x) \cdot 5}{8}$ .

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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 $f^{- 1} (\frac{\tan (x) \cdot 5}{8})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{\tan (x) \cdot 5}{8}) = \arctan (\frac{8 (\frac{\tan (x) \cdot 5}{8})}{5})$

Step 4.2.3

Combine $8$ and $\frac{\tan (x) \cdot 5}{8}$ .

$f^{- 1} (\frac{\tan (x) \cdot 5}{8}) = \arctan (\frac{\frac{8 (\tan (x) \cdot 5)}{8}}{5})$

Step 4.2.4

Multiply $5$ by $8$ .

$f^{- 1} (\frac{\tan (x) \cdot 5}{8}) = \arctan (\frac{\frac{40 \tan (x)}{8}}{5})$

Step 4.2.5

Reduce the expression by cancelling the common factors.

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Step 4.2.5.1

Reduce the expression $\frac{40 \tan (x)}{8}$ by cancelling the common factors.

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Step 4.2.5.1.1

Factor $8$ out of $40 \tan (x)$ .

$f^{- 1} (\frac{\tan (x) \cdot 5}{8}) = \arctan (\frac{\frac{8 (5 \tan (x))}{8}}{5})$

Step 4.2.5.1.2

Factor $8$ out of $8$ .

$f^{- 1} (\frac{\tan (x) \cdot 5}{8}) = \arctan (\frac{\frac{8 (5 \tan (x))}{8 (1)}}{5})$

Step 4.2.5.1.3

Cancel the common factor.

$f^{- 1} (\frac{\tan (x) \cdot 5}{8}) = \arctan (\frac{\frac{8 (5 \tan (x))}{8 \cdot 1}}{5})$

Step 4.2.5.1.4

Rewrite the expression.

$f^{- 1} (\frac{\tan (x) \cdot 5}{8}) = \arctan (\frac{\frac{5 \tan (x)}{1}}{5})$

Step 4.2.5.2

Divide $5 \tan (x)$ by $1$ .

$f^{- 1} (\frac{\tan (x) \cdot 5}{8}) = \arctan (\frac{5 \tan (x)}{5})$

Step 4.2.6

Cancel the common factor of $5$ .

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Step 4.2.6.1

Cancel the common factor.

$f^{- 1} (\frac{\tan (x) \cdot 5}{8}) = \arctan (\frac{5 \tan (x)}{5})$

Step 4.2.6.2

Divide $\tan (x)$ by $1$ .

$f^{- 1} (\frac{\tan (x) \cdot 5}{8}) = \arctan (\tan (x))$

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 $f (\arctan (\frac{8 x}{5}))$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\arctan (\frac{8 x}{5})) = \frac{\tan (\arctan (\frac{8 x}{5})) \cdot 5}{8}$

Step 4.3.3

The functions tangent and arctangent are inverses.

$f (\arctan (\frac{8 x}{5})) = \frac{\frac{8 x}{5} \cdot 5}{8}$

Step 4.3.4

Combine $\frac{8 x}{5}$ and $5$ .

$f (\arctan (\frac{8 x}{5})) = \frac{\frac{8 x \cdot 5}{5}}{8}$

Step 4.3.5

Multiply $5$ by $8$ .

$f (\arctan (\frac{8 x}{5})) = \frac{\frac{40 x}{5}}{8}$

Step 4.3.6

Reduce the expression by cancelling the common factors.

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Step 4.3.6.1

Reduce the expression $\frac{40 x}{5}$ by cancelling the common factors.

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Step 4.3.6.1.1

Factor $5$ out of $40 x$ .

$f (\arctan (\frac{8 x}{5})) = \frac{\frac{5 (8 x)}{5}}{8}$

Step 4.3.6.1.2

Factor $5$ out of $5$ .

$f (\arctan (\frac{8 x}{5})) = \frac{\frac{5 (8 x)}{5 (1)}}{8}$

Step 4.3.6.1.3

Cancel the common factor.

$f (\arctan (\frac{8 x}{5})) = \frac{\frac{5 (8 x)}{5 \cdot 1}}{8}$

Step 4.3.6.1.4

Rewrite the expression.

$f (\arctan (\frac{8 x}{5})) = \frac{\frac{8 x}{1}}{8}$

Step 4.3.6.2

Divide $8 x$ by $1$ .

$f (\arctan (\frac{8 x}{5})) = \frac{8 x}{8}$

Step 4.3.7

Cancel the common factor of $8$ .

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Step 4.3.7.1

Cancel the common factor.

$f (\arctan (\frac{8 x}{5})) = \frac{8 x}{8}$

Step 4.3.7.2

Divide $x$ by $1$ .

$f (\arctan (\frac{8 x}{5})) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \arctan (\frac{8 x}{5})$ is the inverse of $f (x) = \frac{\tan (x) \cdot 5}{8}$ .

$f^{- 1} (x) = \arctan (\frac{8 x}{5})$