Thread: Finding a composite inverse function.

Hey guys! I need to find:



Here's how I'm going about finding the answer:

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:

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!

Originally Posted by JennyFlowers

Hey guys! I need to find:



Here's how I'm going about finding the answer:

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:

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,
returns you an angle

so,

and

Out of curiosity, what difference would this make?



Does switching which function is the inverse change the outcome?

Thanks for the help!

Hello, JennyFlowers!

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

Find: .

Here's how I'm going about finding the answer:

I read this as: "Tangent of the angle whose cotangent is " . Good!

Then: .

The right triangle looks like this:

Code:

                  *
                * |
              *   |  _
            *     | √3
          *       |
        * θ       |
      * - - - - - *
           -1

And we see that: .

. . Hence: .



Do you see what we've done?

We've shown that: .

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



So we can replace with

And the original problem becomes: .

. . which equals (of course): .


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


**

The same reasoning that allows us to say:

. .

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?

"Tangent" is periodic with period . Drawing your angle in quadrant IV rather than quadrant II was just adding [itex]\pi[/itex] radians to it.

In any case, since tan(x)= 1/cot(x), .