The Easiest Way to Determine Trigonometric Factors in Physics EVAR

1
The Easiest Way to Determine Trigonometric
Factors in Physics EVAR!
Vectors play an integral part in physics, whether
they represent a particular force at a point
(such as electric, gravitational, magnetic, or
normal), a relative velocity, or a generalized
vector field representing a force or perhaps even
a vector potential field. Much of the
understanding of physical phenomena in the world
involves being able to understand what the nature
of vector field is created by it. In many cases
we need to know how a vector interacts with
another vector or field, and for this we often
need to extract important information about the
vector. In particular, we often (almost in all
cases) need to know components of the vector in
question relative to a coordinate system that we
have set. Most likely this coordinate system is
not the standard one that we are used to. For the
mathematically inclined, it will be recognized
that this idea is simply finding the projection
of a vector onto an arbitrary axis.
Unfortunately, the axes were using are often
times aligned unusually and the angle might not
even be given explicitly. There is an easy way to
find the projection, though. Whats best, is you
dont have to twist your neck around looking at
angles.
First, lets say you are given a vector and you
know or can figure out the angle it makes with
the origin, as in Figure a. You need to know, for
some reason, (for example you might be balancing
out forces in the x-direction) its x-component.
Well, thats easy of course because everything is
oriented the way weve seen in hundreds of math
and physics books. As shown in the figure, the
x-component has a cos(q) factor and the
y-component has a sin(q) factor.
Now lets say that instead of the ordinary
coordinate system, we need to figure out
components of a vector along the coordinate
system in Figure b. Such a coordinate system
might arise if were looking at a blocks motion
on an incline subject to a pulley. A sample
vector is drawn in Figure c. Now you might say
that finding the components is easy and youd
be right, it is. If you dont have any trouble
with this, you can use this method as a check,
but most students of mechanics stumble a little
at first on the projections of vectors and how to
visualize them.
To find the trigonometric factors that go with
which components, simply imagine the angle at an
extreme case (even though it might not be in
reality). That is, make the angle either very
small or very large. In figure d, what have made
the angle very large (almost 90 degrees). Now
what is the projection of our vector on the
x-axis. Well, since we see that it has to be very
small (approaching 0 as we approach 90 degrees)
we ask what trig factor is 0 at 90 degrees, and
we immediately get the answer cos(q). As a
check, we can look at figure e, making the angle
very close to 0. This checks our answer, because
we know that a cosine of a very small angle is
close to the original value, as we intuitively
see.