Interesting applications of radians To give some background, I am currently a graduate student with a teaching assistantship and am teaching precalculus this semester. The department really pushes us to hammer the differences and similarities between degrees and radians and instances where you might use one over the other and so I am looking for some nice applications of both degrees and radians, but especially radians since most students are already comfortable with degrees. One of the main points the department likes us to make is that calculating arc length is much easier when working with radians as opposed to degrees, but I would like some additional examples. 
I am at a larger state school and the audience is primarily first year college students or students taking math again for the first time in a couple of years, so ideally examples shouldn't be esoteric and require a lot of knowledge of mathematics; so things such as $$ \lim_{h \rightarrow 0} \frac{\sin(h)}{h} = 1$$
are not what I am looking for. Bonus points for examples where working with radians as opposed to degrees makes your life simpler. 
 A: The real reasons mathematicians like radians so much are largely
due to calculus. It's hard to completely ignore that.
On the other hand, some of what calculus does is to prove
certain facts about angles in radians that are easily
observed even with beginning precalculus skills.
For example, have the students put their calculators in to "radians" mode
and have them calculate the sines and tangents of some not-too-large
angles such as $0.1,$ $0.02,$ $0.005,$ and $0.001.$
Point out how close the answers are to the inputs
(in my example, $0.1,$ $0.02,$ $0.005,$ and $0.001,$ respectively).
Now ask each student to imagine that for $20\%$ of the grade in this course,
he, she, or they will have five seconds to give the sine or tangent
of an angle as a decimal number, accurate to within one percent of the
exact value.
The angle is guaranteed to be less than $5$ degrees.
Would the student prefer to receive the value of the angle in degrees
or in radians?
This example exploits some facts the students may later learn in
calculus: the rapid convergence of the Taylor series for
the sine and tangent functions, as well as the rapid convergence
of the limits in
$$ \lim_{h \rightarrow 0} \frac{\sin(h)}{h} = 
\lim_{h \rightarrow 0} \frac{\tan(h)}{h} = 1.$$
But the students do not need to know those facts now in order to
observe how nicely a small-angle measurement in radians corresponds
to the sine or tangent of the same angle.
Another trick you can show them, which takes just a little more work
on their part, is to use $1 - \frac12x^2$ as an approximation for $\cos x$.
For example, the approximation $1 - \frac12(0.1)^2 = 0.995$ agrees
with the exact value of $\cos(0.1)$ to five decimal places.
An application of the small-angle approximation of $\sin x$
is demonstrated by the method for measuring the distance to a star,
as explained in an answer to a similar question.
And if any of the students have an interest in computer programming,
point out that the trig functions in most math libraries in software 
require angles to be given in radians.
Admittedly this is somewhat question-begging: why do writers of software
libraries prefer radians? You can try to explain this
(the reasons mostly come back to why mathematicians prefer radians) 
or simply let the fact that these students will want to use these libraries
be a motivation to get comfortable with radians.
