Expert: Josh - 2/23/2009

Question
I ask John he said it was out of his expertise. Hope you can help, truly needed this answered before monday evening central time.

Question:  People who believe in biorhythms clam that there are three cycles that rule our behavior-the physical,emotional, and mental. Each is a sine functions of a certain period. The function for our emotional fluctuations is:

                     E = sin pi/14 t

where t is measured in days starting at birth. Emotional fluctuations,E,are measured from -1 to 1,inclusive,with 1 representing peak emotional well-being,-1 representing the low for emotional well-being, and 0 representing feeling neither emotionally high nor low.

1. Find E corresponding to t = 7,14,21,28, and 35. Describe what you observe.

2. What is the period of emotional cycle in days?

Answer
It's easier to answer Q2 first.

Some basic intuition about emotion fluctuation:

Consider the expression E=sin(pi*t/14). The sinusoidal function always repeats itself every 2*pi radian. i.e., sin((pi/14)*t)=sin((pi/14)*t+2*pi*n) for any integer n. Thus, with n=1, we get a fundamental period of T=(2*pi)/(pi/14)=28 days. Surprise, surprise, this roughly coincides with the monthly cycle on the lunar calender.

Answer to Q1:
We can plug these numbers into a calculator and evaluate E(t)=sin((pi/14)*t) at t=7142128 and 35 respectively.
Alternatively, we can use the periodic property, divide 7142128 by 28 and find its remainder to be zero.

Hence, E(7142128) has the same value as E(0) which equals sin(0)=0.

E(35)=sin((35/14)*pi)
    =sin(2.5*pi)
    =sin(pi/2) ...since function is 2*pi-periodic
    =1

These days show neutral emotion and peak positivity.

Don't blame me if my response doesn't reach you in time. I live in Australia, a completely different time zone. Cheers:)