Trigonometry Examples

$\cos (\arccos (u))$

Step 1

Interchange the variables.

$u = \cos (\arccos (y))$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $\cos (\arccos (y)) = u$ .

$\cos (\arccos (y)) = u$

Step 2.2

Take the inverse cosine of both sides of the equation to extract $\arccos (y)$ from inside the cosine.

$\arccos (y) = \arccos (u)$

Step 2.3

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

$y = \cos (\arccos (u))$

Step 2.4

Simplify the right side.

Tap for more steps...

Step 2.4.1

The functions cosine and arccosine are inverses.

$y = u$

Step 3

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

$f^{- 1} (u) = u$

Step 4

Verify if $f^{- 1} (u) = u$ is the inverse of $f (u) = \cos (\arccos (u))$ .

Tap for more steps...

Step 4.1

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

Step 4.2

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

Tap for more steps...

Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (\cos (\arccos (u)))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\cos (\arccos (u))) = \cos (\arccos (u))$

Step 4.2.3

Remove parentheses.

$f^{- 1} (\cos (\arccos (u))) = \cos (\arccos (u))$

Step 4.2.4

The functions cosine and arccosine are inverses.

$f^{- 1} (\cos (\arccos (u))) = u$

Step 4.3

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

Tap for more steps...

Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (u)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (u) = \cos (\arccos (u))$

Step 4.3.3

The functions cosine and arccosine are inverses.

$f (u) = u$

Step 4.4

Since $f^{- 1} (f (u)) = u$ and $f (f^{- 1} (u)) = u$ , then $f^{- 1} (u) = u$ is the inverse of $f (u) = \cos (\arccos (u))$ .

$f^{- 1} (u) = u$