Question 549783
<font face="Times New Roman" size="+2">


The tree you are barking up is not only the wrong tree, it is in the totally wrong part of the forest.


Let's begin again.


The periodicity of cotangent is *[tex \Large \pi], hence *[tex \Large \cot\left(\varphi\ +\ k\pi\right)\ =\ \cot\left(\varphi\right)] where *[tex \Large k\ \in\ \mathbb{Z}].  So let *[tex \Large k\ =\ 6], then:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \cot\left(-\frac{17\pi}{3}\ +\ \frac{18}{3}\pi\right)\ =\ \cot\left(\frac{\pi}{3}\right)]


We also know that


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \cot\left(\vartheta\right)\ =\ \frac{\cos\left(\vartheta\right)}{\sin\left(\vartheta\right)}]


Then, looking at the unit circle:


<img src="http://www.math.ucsd.edu/~jarmel/math4c/Unit_Circle_Angles.png">


where we recall that the value of *[tex \Large cos] is the *[tex \Large x]-coordinate of the intersection of the terminal ray of the angle and the unit circle and the *[tex \Large \sin] is the *[tex \Large y]-coordinate of the same point, we can determine:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \cos\left(\frac{\pi}{3}\ =\ \frac{1}{2}\right)]


and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sin\left(\frac{\pi}{3}\ =\ \frac{\sqrt{3}}{2}\right)]


Divide and rationalize the denominator to obtain:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \cot\left(-\frac{17\pi}{3}\right)\ =\ \cot\left(\frac{\pi}{3}\right)\ =\ \frac{\cos\left(\frac{\pi}{3}\right)}{\sin\left(\frac{\pi}{3}\right)}\ =\ \frac{\sqrt{3}}{3}]


One more thing.  "pie" is a prepared food item consisting, generally, of a more or less flakey crust filled with some sort of sweet, often fruit based, filling in the case of pies created as a dessert course, or savory filling in the case of those created as either a main or side dish.  "pi" is the romanization of the lower case Greek alphabet character *[tex \Large \pi] representing the ratio of a circle's circumference to its diameter.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
<div style="text-align:center"><a href="http://outcampaign.org/" target="_blank"><img src="http://cdn.cloudfiles.mosso.com/c116811/scarlet_A.png" border="0" alt="The Out Campaign: Scarlet Letter of Atheism" width="143" height="122" /></a></div>
</font>