Trigonometry Examples

$f (x) = \frac{1}{2} \tan (x)$

Step 1

Write $f (x) = \frac{1}{2} \tan (x)$ as an equation.

$y = \frac{1}{2} \tan (x)$

Step 2

Interchange the variables.

$x = \frac{1}{2} \cdot \tan (y)$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $\frac{1}{2} \cdot \tan (y) = x$ .

$\frac{1}{2} \cdot \tan (y) = x$

Step 3.2

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \tan (y)) = 2 x$

Step 3.3

Simplify the left side.

Tap for more steps...

Step 3.3.1

Simplify $2 (\frac{1}{2} \tan (y))$ .

Tap for more steps...

Step 3.3.1.1

Combine $\frac{1}{2}$ and $\tan (y)$ .

$2 \frac{\tan (y)}{2} = 2 x$

Step 3.3.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.1.2.1

Cancel the common factor.

$2 \frac{\tan (y)}{2} = 2 x$

Step 3.3.1.2.2

Rewrite the expression.

$\tan (y) = 2 x$

Step 3.4

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

$y = \arctan (2 x)$

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \arctan (2 x)$ is the inverse of $f (x) = \frac{1}{2} \cdot \tan (x)$ .

Tap for more steps...

Step 5.1

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

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

$f^{- 1} (\frac{1}{2} \cdot \tan (x)) = \arctan (2 (\frac{1}{2} \cdot \tan (x)))$

Step 5.2.3

Combine $\frac{1}{2}$ and $\tan (x)$ .

$f^{- 1} (\frac{1}{2} \cdot \tan (x)) = \arctan (2 (\frac{\tan (x)}{2}))$

Step 5.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.4.1

Cancel the common factor.

$f^{- 1} (\frac{1}{2} \cdot \tan (x)) = \arctan (2 (\frac{\tan (x)}{2}))$

Step 5.2.4.2

Rewrite the expression.

$f^{- 1} (\frac{1}{2} \cdot \tan (x)) = \arctan (\tan (x))$

Step 5.3

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

Tap for more steps...

Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

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

$f (\arctan (2 x)) = \frac{1}{2} \cdot \tan (\arctan (2 x))$

Step 5.3.3

The functions tangent and arctangent are inverses.

$f (\arctan (2 x)) = \frac{1}{2} \cdot (2 x)$

Step 5.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.3.4.1

Factor $2$ out of $2 x$ .

$f (\arctan (2 x)) = \frac{1}{2} \cdot (2 (x))$

Step 5.3.4.2

Cancel the common factor.

$f (\arctan (2 x)) = \frac{1}{2} \cdot (2 x)$

Step 5.3.4.3

Rewrite the expression.

$f (\arctan (2 x)) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \arctan (2 x)$ is the inverse of $f (x) = \frac{1}{2} \cdot \tan (x)$ .

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