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.
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.