Yes, some of my college students have the same problem. Some of the
problem goes away if you can just get the students to sketch the vector
on an x-y coordinate system. But their insistance on drawing a right
triangle is also part of the problem. In more detail...
Part of the problem stems from the fact that students tend to view the
trig relationships from a right triangle perspective. If the vector is
-3i + 4j they are going to draw a right triangle in quadrant II and
solve for the angle of that triangle. Some of them won't even get that
correct... they'll invert the 4/3 and find the arctan of 3/4. Some will
include the minus sign and some won't. So we might get any of the
following answers... ( 53.1, -53.1, 36.9, -36.9 )... when the correct
answer is 126.9 degrees.
Another part of the problem is their calculators cannot tell the
difference between quadrants II and IV, nor between quadrants I and III.
Therefore, even if they correctly enter arctan(-4/3) into their
calculator, they have to realize the need to change the -53.1 calculator
result into 126.9. Even if they do realize this, some will add the 53.1
to 90, coming up with 143.1, rather than adding the -53.1 to 180 to get
the correct 126.9.
Below are the things I tell my students. If they pay attention and do
what I tell them, it helps... but some cannot avoid drawing the triangle
because the only way they view trig relationships is via SOHCAHTOA.
That is, if they can't use the acronym they've memorized, they're lost.
(1) Sketch the vector on an x-y coordinate system. This is only a
sketch, not a graph, but do it accurately enough that you can tell not
only what quadrant the vector is in, but also whether it is more or less
than half-way through that quadrant.
(2) DO NOT DRAW A TRIANGLE.
(3) Use the arctan key on your calculator and always enter the j-part
over the i-part... angle = arctan[(j-part)/(i-part)]. Be sure to
include any minus signs for both the i-part and j-part.
(4) If your vector is in quadrant I or IV, you can report the angle the
calculator gives you. If your vector is in quadrant II or quadrant III
then you have to add 180 degrees to whatever the calculator gives you.
(5) After writing down the angle you just calculated, look at your
sketch and see if the sketch and calculated angle agree. If so,
great... if not, start over and see where you went wrong.
These steps will always work if the students would just do them.
Getting them to sketch the vector is the hardest part. Getting them to
refrain from drawing a triangle is the next hardest part.
Michael D. Edmiston, Ph.D.
Professor of Chemistry and Physics
Bluffton College
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu