# Thread: Finding a composite inverse function.

1. ## Finding a composite inverse function.

Hey guys! I need to find:

$\displaystyle tan(cot^{-1}(\frac{-1}{\sqrt{3}}))$

I read this as, "Tangent of the angle whose cotangent is negative 1 over the square root of 3."

As I understand it, I need to draw the triangle for the inverse cotangent in quadrant 4 since it is negative. Here's my drawing:

Now, using this drawing, I would find the tangent (and the answer to the expression) to be: $\displaystyle -\sqrt{3}$

Did I do this correctly? I'm having a hard time with the concept of inverse functions in general, so any advice or somewhere I could read to better understand the concept would be helpful. I've read my trig book and it's not helping me very much.

I'm doubting that it is correct because when I evaluate the expression on my TI-84, the answer doesn't match.

Thanks for any help!

2. Originally Posted by JennyFlowers
Hey guys! I need to find:

$\displaystyle tan(cot^{-1}(\frac{-1}{\sqrt{3}}))$

I read this as, "Tangent of the angle whose cotangent is negative 1 over the square root of 3."

As I understand it, I need to draw the triangle for the inverse cotangent in quadrant 4 since it is negative. Here's my drawing:

Now, using this drawing, I would find the tangent (and the answer to the expression) to be: $\displaystyle -\sqrt{3}$

Did I do this correctly? I'm having a hard time with the concept of inverse functions in general, so any advice or somewhere I could read to better understand the concept would be helpful. I've read my trig book and it's not helping me very much.

I'm doubting that it is correct because when I evaluate the expression on my TI-84, the answer doesn't match.

Thanks for any help!
Actually,
$\displaystyle cot^{-1}(\frac{-1}{\sqrt 3})$ returns you an angle

so, $\displaystyle cot^{-1}(\frac{-1}{\sqrt 3}) = \frac{-\pi}{3}$

and $\displaystyle tan(\frac{-\pi}{3})= -tan(\frac{\pi}{3})= -\sqrt{3}$

$\displaystyle \therefore tan(cot^{-1}(\frac{-1}{\sqrt 3})) = -\sqrt{3}$

3. Out of curiosity, what difference would this make?

$\displaystyle tan^{-1}(cot(\frac{-1}{\sqrt{3}}))$

Does switching which function is the inverse change the outcome?

Thanks for the help!

4. Hello, JennyFlowers!

With a little thought, this is an "eyeball" problem.
(You get the answer by looking at the problem.)

Find: .$\displaystyle \tan\left[\cot^{-1}\left(-\tfrac{1}{\sqrt{3}}\right)\right]$

I read this as: "Tangent of the angle whose cotangent is $\displaystyle -\tfrac{1}{\sqrt{3}}$ " . Good!

$\displaystyle \text{Let }\theta \,=\,\cot^{-1}\left(-\tfrac{1}{\sqrt{3}}\right) \quad \hdots \quad \theta\text{ is an angle whose cotangent is }-\tfrac{1}{\sqrt{3}}$

Then: .$\displaystyle \cot\theta \:=\:-\frac{1}{\sqrt{3}} \:=\:\frac{adj}{opp}$

The right triangle looks like this:
Code:
                  *
* |
*   |  _
*     | √3
*       |
* θ       |
* - - - - - *
-1

And we see that: .$\displaystyle \tan\theta \:=\:\frac{\sqrt{3}}{\text{-}1} \:=\:\text{-}\sqrt{3}$

. . Hence: .$\displaystyle \theta \:=\:\tan^{-1}\left(\text{-}\sqrt{3}\right)$

Do you see what we've done?

We've shown that: .$\displaystyle \cot^{-1}\left(\text{-}\tfrac{1}{\sqrt{3}}\right) \;=\;\tan^{-1}\left(\text{-}\sqrt{3}\right)$

Give it some thought . . . The reasoning should become clear. .**

So we can replace $\displaystyle \cot^{-1}\left(\text{-}\tfrac{1}{\sqrt{3}}\right)$ with $\displaystyle \tan^{-1}\left(\text{-}\sqrt{3}\right)$

And the original problem becomes: .$\displaystyle \tan\left(\tan^{-1}(\text{-}\sqrt{3})\right)$

. . which equals (of course): .$\displaystyle -\sqrt{3}$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

**

The same reasoning that allows us to say:

. . $\displaystyle \begin{array}{ccc}\sec^{-1}\left(\dfrac{2}{\sqrt{3}}\right) &=& \cos^{-1}\left(\dfrac{\sqrt{3}}{2}\right) \\ \\[-2mm] \csc^{-1}(\sqrt{2}) &=& \sin^{-1}\left(\dfrac{1}{\sqrt{2}}\right) \end{array}$

5. Thank you, Soroban, I think I understand now. But how did you determine that the angle should be drawn in quadrant II? Was it just luck that I got the same answer by drawing in quadrant IV?

6. "Tangent" is periodic with period $\displaystyle \pi$. Drawing your angle in quadrant IV rather than quadrant II was just adding $\pi$ radians to it.

In any case, since tan(x)= 1/cot(x), $\displaystyle tan(cot^{-1}(-\frac{1}{\sqrt{3}})= 1/cot(cot^{-1}(\frac{1}{\sqrt{3}}))= -1/\frac{1}{\sqrt{3}}= -\sqrt{3}$.