Trigonometry Examples

cot (x)

$\cot (x)$

Step 1

Interchange the variables.

x = cot (y)

$x = \cot (y)$

Step 2

Solve for

y

$y$ .

Tap for more steps...

Step 2.1

Rewrite the equation as

cot (y) = x

$\cot (y) = x$ .

cot (y) = x

$\cot (y) = x$

Step 2.2

Take the inverse cotangent of both sides of the equation to extract

y

$y$ from inside the cotangent.

y = arccot (x)

$y = arccot (x)$

Step 2.3

Remove parentheses.

y = arccot (x)

$y = arccot (x)$

y = arccot (x)

$y = arccot (x)$

Step 3

Replace

y

$y$ with

f^{- 1} (x)

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

f^{- 1} (x) = arccot (x)

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

Step 4

Verify if

f^{- 1} (x) = arccot (x)

$f^{- 1} (x) = arccot (x)$ is the inverse of

f (x) = cot (x)

$f (x) = \cot (x)$ .

Tap for more steps...

Step 4.1

To verify the inverse, check if

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

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

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

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

Step 4.2

Evaluate

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

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

Tap for more steps...

Step 4.2.1

Set up the composite result function.

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

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

Step 4.2.2

Evaluate

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

$f^{- 1} (\cot (x))$ by substituting in the value of

f

$f$ into

f^{- 1}

$f^{- 1}$ .

f^{- 1} (cot (x)) = arccot (cot (x))

$f^{- 1} (\cot (x)) = arccot (\cot (x))$

f^{- 1} (cot (x)) = arccot (cot (x))

$f^{- 1} (\cot (x)) = arccot (\cot (x))$

Step 4.3

Evaluate

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

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

Tap for more steps...

Step 4.3.1

Set up the composite result function.

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

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

Step 4.3.2

Evaluate

f (arccot (x))

$f (arccot (x))$ by substituting in the value of

f^{- 1}

$f^{- 1}$ into

f

$f$ .

f (arccot (x)) = cot (arccot (x))

$f (arccot (x)) = \cot (arccot (x))$

Step 4.3.3

The functions cotangent and arccotangent are inverses.

f (arccot (x)) = x

$f (arccot (x)) = x$

f (arccot (x)) = x

$f (arccot (x)) = x$

Step 4.4

Since

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

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

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

$f (f^{- 1} (x)) = x$ , then

$f^{- 1} (x) = arccot (x)$ is the inverse of

$f (x) = \cot (x)$ .

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