# Intersections of Sine Graphs

1. Nov 2, 2013

### Trav44

1. The problem statement, all variables and given/known data

Hi guys I was wondering how to find the points of intersections between 3 different sine graphs.

For an assignment I am trying to find when my three biorhythms (Physical, Emotional, Intellectual) will all cross at once.

Each cycle runs on the following time frame:

Physical Cycle - 23 days

Emotional Cycle - 28 days

Intellectual Cycle - 33 days

2. Relevant equations

I have converted the three into equations

Physical
A=100 for 100%
B=
T=2pi/B which can be re-arranged to B=2pi/T
B=2pi/23
B=0.2732
C=N/A
D=N/A
y=100sin0.2732x

Emotional
A=100 for 100%
B=
T=2pi/B which can be re-arranged to B=2pi/T
B=2pi/28
B=0.2244
C=N/A
D=N/A
y=100sin0.2244x

Intellectual
A=100 for 100%
B=
T=2pi/B which can be re-arranged to B=2pi/T
B=2pi/33
B=0.1904
C=N/A
D=N/A
y=100sin0.1904x

Am I calculating the equation for each of these graphs correctly?

How do I find when all of the graphs will equal the same x-value at one time?

3. The attempt at a solution

I have no idea where I would start? Is there an equation for this? or do i put a certain number as the x-value for each graph? Find factors of the periods multiplied together? Something else?

Any help will be much appreciated thank you :)

2. Nov 2, 2013

### LCKurtz

I suppose you mean have the same y values for some value of x? Your three sines all start at 0 and they will all be 0 again at the common period, which is the lowest common multiple of their periods: 23*28*33 = 21252 days = 58 years. Of course that's assuming all three traits are at zero to start with. And do negative biorhythms make sense?

I doubt the three curves cross at any other time, but I'm just guessing. But, then, I don't know anything about biorhythms, such as, for example, are they really periodic?

3. Nov 2, 2013

### Trav44

Thanks for the quick reply but that's not exactly what I was looking for :/

Guess I should have explained what biorhythms are.

They are charts that illustrate the principle that we are influenced by physical, emotional, and intellectual cycles. Many people report that they can improve the quality of their lives by monitoring the highs and lows of these cycles and acting accordingly.

They start from your day of birth so their starting point on the x-axis is different for everyone.
The x-axis is a measure of time in days represented by dates

They are 100% periodic

When two of the lines cross it is called a 'critical day'
When three lines cross it is called a 'double critical day'

I am trying to calculate when these critical days occur.

Below are 2 example pictures of biorhythms:

One with just the biorhythm sine graphs
and one indicating what a critical day is (RED BOX).

Thanks for any help :)

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4. Nov 2, 2013

### LCKurtz

If you want to see when two out of phase sine waves cross each other you need to solve an equation in the form $\sin x =\sin(x-a) = \sin x \cos a - \cos x \sin a$. This can be expressed as$$\tan x = \frac{-\sin a}{1-\cos a}$$which can be solved numerically. I don't care to go into more detail with this problem because, apparently, biorhythms are controversial in the science community, and it would seem like a waste my time. Good luck with your project.