This is a fairly short section, but contains some errors
that I see my calculus students continually making so I thought I’d include
them here as a separate section.
Degrees vs. Radians
Most trig classes that I’ve seen taught tend to concentrate
on doing things in degrees. I suppose
that this is because it’s easier for the students to visualize, but the reality
is that almost all of calculus is done in radians and students too often come
out of a trig class ill prepared to deal with all the radians in a calculus
You simply must get used to doing everything in radians in a
calculus class. If you are asked to
evaluate at we are asking you to use 10 radians not 10
degrees! The answers are very, very
different! Consider the following,
You’ll notice that they aren’t even the same sign!
So, be careful and make sure that you always use radians
when dealing with trig functions in a trig class. Make sure your calculator is set to
calculations in radians.
I see students attempting both of the following on a
These just simply aren’t true. The only reason that I can think of for these
mistakes is that students must be thinking of as a multiplication of something called cos and x. This couldn’t be
farther from the truth! Cosine is a
function and the cos is used to
denote that we are dealing with the cosine function!
If you’re not sure you believe that those aren’t true just
pick a couple of values for x and y and plug into the first example.
So, it’s clear that the first isn’t true and we could do a
similar test for the second example.
I suppose that the problem is that occasionally there are
values for these that are true. For
instance, you could use in the second example and both sides would be
zero and so it would work for that value of x. In general however, for the vast majority of
values out there in the world these simply aren’t true!
On a more general note.
I picked on cosine for this example, but I could have used any of the
six trig functions, so be careful!
Powers of trig functions
Remember that if n
is a positive integer then
The same holds for all the other trig functions as well of
course. This is just a notational
idiosyncrasy that you’ve got to get used to.
Also remember to keep the following straight.
In the first case we taking the tangent then squaring result
and in the second we are squaring the x then taking the tangent.
The is actually not the best notation for this
type of problem, but I see people (both students and instructors) using it all
the time. We really should probably use
to make things clear.
Inverse trig notation
The notation for inverse trig functions is not the best. You need to remember, that despite what I
just got done talking about above,
This is why I said that n was a positive integer in the
previous discussion. I wanted to avoid
this notational problem. The -1 in is NOT an exponent, it is there to denote the
fact that we are dealing with an inverse trig function.
There is another notation for inverse trig functions that
avoids this problem, but it is not always used.