# How to find intersection of two and three sine waves on x axis intercept? (Biorhythms)

I'm currently studying the trigonometry behind biorhythms. I was reading through the Wikipedia article on the topic (https://en.wikipedia.org/wiki/Biorhythm) which states that:

Basic arithmetic shows that the combination of the simpler 23- and 28-day cycles repeats every 644 days (or 1-3/4 years), while the triple combination of 23-, 28-, and 33-day cycles repeats every 21,252 days (or 58.18+ years).

From my understanding, these are instances where they intersect on the x axis, but I may be mistaken.

What I am looking for are the points in time when two and three of the sine waves intersect and do so exactly when intercepting the x axis (x, 0). These are refered to as "double critical" and "super critical" days respectively. I have read articles and watched videos explaining how to find intercepts of sine waves, but I am unable to find an explaination as to how this can be used when a particular coordinate is wanting to be found.

The equations for the three waves are as follows:

• Physical: $\sin(2πx/23)$
• Emotional: $\sin(2πx/28)$
• Intellectual: $\sin(2πx/33)$

All the waves start at the point (0,0) at the date of birth. For consistancy, let's make that 1/1/1970.

Would someone be able to explain the process required to solve this?

Thank you in advance,

Lachlan

• You might be interested in the wiki articles on BioRhtyhms & on Pseudo-Science or Martin Gardner's essay "Freud, Fleiss, and BioRhythms". – DanielWainfleet Sep 1 '18 at 20:05

## 1 Answer

To determine how many days it takes for the cycle to repeat itself, multiply the different cycles by themselves.

For example:

23 × 28 = 644 days

Finding the intersection point requires finding the LCM of each combination.

• No. The right way is to take the least common multiple of the lengths. The LCM of 22 and 28 is 308. The values in the Q are pair-wise co-prime so the LCMs are the products. – DanielWainfleet Sep 1 '18 at 20:01