Proof that the range of the cotangent function is all reals

[jsh_aziza]

Could someone please help me with this equation:

Prove that the range of the cotangent function is the set of all real numbers. (Hint: Use the unit circle and the point (x,y) that corresponds to cot theta)

I really hate proofs because I know the ultimate answer but it is hard for me to go about using the different theorems and steps in getting there, so if anyone can help through those steps, I would really appreciate it. So far what I have is:

cot theta = x/y

Let cot theta = x/y = a, so then x = ay. Using the unit circle, note that x^2 + y^2 = 1. Using substitution, we get:

a^2 y^2 + y^2 = 1

y^2(a^2 + 1) = 1

y^2 = 1 / (a^2 + 1)

y = +/- 1 / sqrt(a^2 + 1)

Did I do it right? And if so how would I present it as a proof? Thanks for any help you can give me.

 

[stapel]

Did I do it right?

If you had been asked to prove that y = +/- 1 / sqrt[a[sup:1mnqq1ho]2[/sup:1mnqq1ho] + 1], then you would have done the proof correctly. Unfortunately, this is not what you proved.

Instead, you were asked to prove that, for any real number "w", there is some value of \(\displaystyle \theta\) for which \(\displaystyle \cot{(\theta)}\, =\, w\). This is a very different thing. :shock:

You know that, for any angle \(\displaystyle \theta\), you have, on the unit circle, \(\displaystyle \cot{(\theta)}\, =\, \frac{y}{x}\) for some values of x and y between -1 and 1. Is there any way to relate w to the ratio y/x, and thus to the cotangent? This relation is what you need to show. :wink:

Eliz.