Subject: Trigonometry in Marine Biology

Robert Kenney (
Thu, 10 Dec 1998 18:46:01 -0500

At 13:04 12/10/98 PST, you wrote:
>Hello. My name is Natalie Kunz and I'm a student at Marquand-Zion high 
>school in Missouri.
>For Trigonometry class, we have to do projects on how trig relates to 
>the "real world."
>Since marine biology is way interesting, I decided to see how scientists 
>use sine and cosine curves to help track whales.
>DO scientists use sine curves? 
Hi, Natalie (how did you pick "blondechicken" for your email name?).  The
questions are certainly getting interesting this year.  Let's see, sine
curves and whale-tracking?  

How's this for a stretch - a sound underwater is actually a wave of pressure
that varies with time that follows a sine curve if you can look at it with
the right equipment.  Scientists using the Navy's submarine tracking
stations have been able to follow whales in the middle of the ocean by
listening to their sounds from a room in Virginia.

That may be stretching a point to answer your question, but in fact many
aspects of marine science (or any science) involve trigonometry (and not
just sine functions).  Many of the terms in the equations that describe how
waves form or how currents flow include trigonometric functions (of course I
forgot all of them the day after I took my last exam in physical
oceanography).  Mathematics is the language of science, and no one can be a
successful scientist any more without some understanding of math.  Maybe
biologists might use less math than astrophysicists, but we still need to
know the basics.

Here's what I use trig for most often - studying whales often involves going
out to sea on a ship and working with the data later.  We use a variety of
navigation equipment to tell where we are, and we keep track of that using
latitude and longitude.  Many times I want to know the distance between two
locations (for example, how far did whale number 1163 move from the spot
where we attached the radio tag to where we saw her again three days later).
The formula for calculating the distance (in nautical miles) from point 1 to
point 2 when you know the latitude and longitude of both is:

   60 arcos[sin(lat1) sin(lat2) + cos(lat1) cos(lat2) cos(lon2 - lon1)]

Easy as pie, right?  I actually use that a lot, except I copied it from a
book into a computer program, so I don't have to really know it.  But there
is something much simpler that I also use frequently - when I want to know
something about distances but don't have my computer.  Do you know why
sailors (and airplane pilots, too) use different miles than the ones we use
on land (nautical miles instead of statute miles)?  It's because a nautical
mile really means something - it is 1/5400th of the distance from the
equator to the north pole.  Since that is 90 degrees of latitude and each
degree contains 60 minutes, one nautical mile equals one minute of latitude.
It also equals one minute of longitude, but only at the equator. If you look
at a globe, the longitude lines get closer together the farther you get from
the equator, until they all meet at the poles.  So how long is one minute of
longitude at any place other than the equator?  What function describes how
the longitude lines get closer when you go from a latitude of 0 degrees to a
latitude of 90 degrees?  COSINE!  Cosine(0) = 1.000 and cosine(90) = 0.  The
cosine of the latitude tells you the length of a minute of longitude
anywhere on Earth.  I'm sitting right now in an office at latitude 41
degrees, 29 minutes, 28 seconds North, the cosine of that is 0.7490585109
(approximately), and one minute of longitude here is about three-quarters of
a nautical mile (1,387 meters).

Dr. Bob

 | Robert D. Kenney, Ph.D.              |
 | University of Rhode Island                                        |
 | Graduate School of Oceanography                                   |
 | Bay Campus - Box 41                          TEL:  (401) 874-6664 |
 | Narragansett, RI 02882-1197, U.S.A.          FAX:  (401) 874-6497 |