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Q:

How does $\alpha = 2 \sin^2( \pi/n)$ lead to $\theta_0 = n \pi/2$?

In number theory and modular forms one often finds the formula:
$$\alpha = 2 \sin^2 ( \pi/n)$$
Now, this leads to the LHS of:
$$\theta(z, \alpha) = 2 \sin^2 (\pi z/n) \cos (\pi z)$$
which, for $\theta_0 = n \pi /2$, gives:
$$\theta(z, \alpha) = \cos(n \pi z)$$
But why is this? How does $\alpha$ lead to this particular value of $\theta_0$?

A:

That just means that $\sin^2(\pi/n) = 1/2$.

Unconscious orientation toward goals in living organisms.
Functional action refers to the ability of an organism to adapt its behavior to its immediate environment in order to optimize its chances of survival and reproduction. The concept of functional action involves the following two elements: 1) a rapid and autonomous scan of its environment to determine the state of affairs, and 2) an efficient behavioral output (i.e., behavior or action) triggered in response to this information, and 3) an accurate readjustment of behavior when circumstances change. Relevant behavior is anticipated and actually prepared for most of the time, even though the organism is temporarily unaware of its promptness. The goal is to allow the organism to continue its trend of evolutionary success. Existing research shows that accurate anticipation of behavior can be achieved by two different functional explanations: 1) “vigilance toward possible dangerous events” and 2) “attention toward potential rewards.” Although both of

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