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In article <47qdri$42b@coranto.ucs.mun.ca>,
Writers Network  <writers@calvin.stemnet.nf.ca> wrote:
>Hi! This is my first time using the net and I have a question.  I was wondering why problems in 
>calculus involving trigonometry must use radian measure instead of degree measure.  Why is this 
>the case?

Unlike the degree, the radian has no dimension.  Why?  An angle of one radian
causes the ratio of the length of the arc of the circle's circumference to
the radius of the circle to be one.   It is clear that dividing the arc length
(units of length) by the radius of the circle (units of length) results in
NO units -- so the radian actually is not a unit!  It's dimensionless.

So why do you have to use radians?  Because the argument of a trigonmetric
function (at least when you take the derivate, or when you integrate it) must
be dimensionless.  Consider the derivative of the function given by

                          f(x) = sin[g(x)]

Where g(x) is any function of x.  Then, by the chain rule,

                         df(x)   d{sin[g(x)]} dg(x)
                         ----- = ------------ -----
                          dx       d[g(x)]     dx

Now consider the case when g(x) represents a constant angle in radians -- a
dimensionless number.   Let's say g(x) = Pi.  Then we have

                         df(Pi)           d(Pi)
                         ------ = cos(Pi) -----
                          dx               dx

Since the derivative of sin is cos.  And since the derivative of a 
constant number is one, clearly we get the result that the derivative of f
evaluated at Pi is simply cos(Pi), or -1.

What happens when the argument of the sin is an angle in degrees, though?
What is the derivative of 180 degrees with respect to x?  It's not a 
dimensionless number, you can't say that its derivative with respect to x
is 1.  That's a simple example of why we have to use radians instead of 
degrees in calculus -- because when we try to integrate or differentiate
trigonometric functions whose arguments are in degrees, things get very
troublesome.

-- 
                 Dave "Bear" Pischke (pischke@ecf.toronto.edu)
___   _____      Engineering is the art of moulding materials we do not fully
|__ |)  |  |  |  understand into shapes we cannot fully analyse and preventing
|__ |\  |  |/\|  the public from realising the full extent of our ignorance.