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## Moving From Right to Wrong to Right

Two weeks ago, attempting to show how times (e.g. goals, content focus, etc.) have changed in mathematics education, I provided a copy of questions from the "1983 Oregon Invitational Mathematics Tournament." Again, I do not know who produced, coordinated, or scored the exam.

One week ago, one reader of this website (T.R., Bellingham) argued that one of the problems had no answer. It was:

[Q#10] How many solutions does the equation cos x = x/(3π) have?
(a) 2 (b) 4 (c) 6 (d) 8 (e) infinitely many

If you look at his associated solution commentary, you will see T.R.'s insight, argument, and nice link to physicist Richard Feynmenn.

But, now C.S. (Mount Vernon) weighs in with another challenge: As usual, I beg to differ about 7 solutions, and the correct answer not being given. The question did not state “real” solutions. Therefore, including complex numbers, there are infinite solutions. The answer WAS given.

To play around with this a little, I created a manipulate that shows the solutions in the complex plane. The grey locator point is x, the red point is cos(x) and the blue point is x/(3pi).

X is a solution when the blue and red dot are on top of each other. Here is one approximate solution.

It’s interesting to watch the movement of the dots fly around the plane for various x values. There are only seven places that they’re on top of each other, and they are all real numbers. Here is where the seven solutions lie (the green points):

However, if the real domain is increased, we see many more solutions in the complex plane.

I’m not even sure if this shows all or not in this domain. I used a pretty rudimentary find root method.

So now the bigger question is...do these solutions lie on parabolas or a hyperbola? Since the cosines of complex numbers are related to hyperbolic functions, my guess is a hyperbola, but I need to experiment more.

Thus, no new question or problem for this week. Rather, I hope you enjoyed T.R.'s and C.S.'s exploration of the problem...and wonder: How did students in 1983 responded to this question?

And, if any other readers have insights to share regarding this problem, please send them to me and I will try to share them.

Hint: Not applicable....