Question

people who believe in biorhythms claim there are three cycles that rule our behavior - the physical, emotional, and mental. each is a sine function of a certain period. the function for our emotional fluctuations is e = sin(\frac{\pi}{20}t) (equation may not be based on actual studies.) where t is measured in days starting at birth. emotional fluctuations, e, are measured from - 1 to 1, with 1 representing peak emotional well - being, - 1 representing the low for emotional well - being, and 0 representing feeling neither emotionally high nor low. a. find e corresponding to t = 50, 60, 70, 80, and 90. describe what you observe. b. what is the period of the emotional cycle?

Explanation:

Step1: Calculate E for t = 50

Substitute t = 50 into $E=\sin\frac{\pi}{20}t$.
$E=\sin\frac{\pi}{20}\times50=\sin\frac{5\pi}{2}= 1$

Step2: Calculate E for t = 60

Substitute t = 60 into $E=\sin\frac{\pi}{20}t$.
$E=\sin\frac{\pi}{20}\times60=\sin3\pi = 0$

Step3: Calculate E for t = 70

Substitute t = 70 into $E=\sin\frac{\pi}{20}t$.
$E=\sin\frac{\pi}{20}\times70=\sin\frac{7\pi}{2}=- 1$

Step4: Calculate E for t = 80

Substitute t = 80 into $E=\sin\frac{\pi}{20}t$.
$E=\sin\frac{\pi}{20}\times80=\sin4\pi = 0$

Step5: Calculate E for t = 90

Substitute t = 90 into $E=\sin\frac{\pi}{20}t$.
$E=\sin\frac{\pi}{20}\times90=\sin\frac{9\pi}{2}=1$
We observe that the values of E cycle between - 1, 0 and 1.

Step6: Find the period of the function

For a sine - function of the form $y = A\sin(Bx - C)+D$, the period is given by $T=\frac{2\pi}{|B|}$.
In the function $E=\sin\frac{\pi}{20}t$, $B = \frac{\pi}{20}$.
$T=\frac{2\pi}{\frac{\pi}{20}}=40$

Answer:

a. When t = 50, E = 1; when t = 60, E = 0; when t = 70, E=-1; when t = 80, E = 0; when t = 90, E = 1. The values of E cycle between - 1, 0 and 1.
b. The period of the emotional cycle is 40 days.