I have been teaching trigonometry for many years and, surprisingly, did not know of this issue. I read the responses and the references therein, and I am putting my thoughts on this issue here.
The question is how to approach trigonometry, right triangles first and then unit circle; or circle first and then triangle.
I think the answer depends on the level of preparation of students, in particular how comfortable are they with the concept of functions when the section on trigonometry starts. At that point we may encounter two category of students, at one extreme are the ones who struggle with basic algebra, and at the other extreme there are students who are comfortable with a question like "Let $f(x)=x^2+3x+7$ for $x \ge -3/2$. Find $f^{-1}$ as well as its domain and range". Does the same book or approach work for both groups? I doubt it. I think my students will do better starting with a brief intro using the triangle approach, polar coordinates, general circle, and then the unit circle.
Just as I would not advocate teaching logarithms in precalculus as the area under a hyperbola, I will also hesitate to start with the unit circle. Unit circle is pretty and offers a clean definition but much of that neatness is lost to my students.
There are decided advantages and disadvantages of triangle approach (T) and circle approach (C). Here are some items I can recall.
1- In (T) the angle takes a back seat to the triangle. The angles are lost in the fray of so many pieces of the triangle. In (C) the focus is on the angle and the function aspect of trig entities comes to light. The student cannot miss the fact that the terminal side of the angle is rotating like the handle of a clock. The angle is prominent from the start, hence the function view becomes central.
2- In (T) you are not tied to a coordinate system, an angle need not be in the standard position, and you do not worry about an orientation for angles. But you are limited to acute angles. In (C) you get the general definition and the find the signs easily.
3- In (T) You may call your angle $x$. Note that this is highly confusing in the unit circle presentation. Most books even avoid writing $y=\cos x$, when they introduce cosine. You are essentially asking students to get accustomed to the following picture. Compare that to how students are used to seeing $(x,y)$ in $y=f(x)$.
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4- In (T) no triangle side is artificially set to be 1. This makes it easier to do some basic trig identities and substitutions. Say, you want to do a trig substitution for $\sqrt{x^2+2x+5}$. This is less error prone for students when you avoid fractions and use a triangle with 'whole' parts.
5- In (C) you can visually approximate trig functions because you set the denominator to be 1. This takes the mystery out of trig functions. Students can eyeball a diagram and have an estimate for a trig function.
6- It is important to emphasize that the input and output of most mathematical functions are dimensionless quantities, and as such, when these functions are used in applications, their input and output are ratios of similar physical quantities. One of the rare examples of this issue that comes up in precalculus is the Richter Scale formula for earthquakes $\dfrac{R}{R_0}=Log \dfrac{E}{E_0}$. It would be attractive to simplify the formula by taking denominators to be 1 but then we risk leaving the wrong impression that one can take the logarithm of energy. In the same fashion, when teaching $y=\cos x$ we need to emphasize that both $x$ and $y$ are ratios. Otherwise students may erroneously think one can take cosine of distance, time, the physical angle itself, or any other physical quantity, and that the output is another physical quantity such as distance. This lack of familiarity with the issue of units can be seen in even advanced math students who take PDEs and evaluate some lengthy integrals related to Fourier series. If they incorrectly come to an answer such as $L+L^3$, where $L$ is supposed to be some length, they do not see the obvious unit mismatch.
