Trigonometric angles of rotation

Question

I find that the notion of trigonometric angles of rotation is a bit confusing for the students. In my curriculum, students learn about angles first time at geometry in middle-school. The angles are "static"(two rays with a common vertex), and have a measure between 0 and 180 degrees. Then in 7th grade the trigonometric functions(Ratios) are introduced in right-triangles and students solve a lot of problems applying them in triangles, and they seem to even like these problems and trig ratios. But then in 9th grade the study of trig functions is continued using the unit circle approach. First of all, the notion of angle is "Extended", we talk about angles greater than 180 degrees and negative angles regarded as "angles of rotation" of a ray(with initial side, terminal side, etc). But i feel these "angles of rotation" are a bit confusing for the students. First of all, because in practical math applications we work with angles greater than 180 very little. With negative angles we don't work at all.(that's the curriculum in my country, it doesn't insist on practical applications, but rather on theoretical ones, it is supposed that students make this practical math in pshysics classes).

Then, there appears some confusion in student's mind between the two types of angles: geometric angle and angle of rotation.For example, when we do the reduction to the first quadrant, we deal with the reference angle, which is somehow no more a rotation angle but rather an ordinary one, because we apply to it the trig ratios in a right triangle For example in this picture:

the 150 angle is a "rotation angle", but the...30 is somehow an ordinary angle. Which may produce confusion from a rigorous point of view. How should we deal with this? Make the 30 also rotation angle by putting an arrow? Also, in this picture:

somehow (-x) is a rotation angle, but at the same time there's an "underlying" ordinary geometrical angle of x which we use when we assert that the two points are symmetrical with respect to the x-axis. So there's again some confusion between two types of angles. How should we eliminate this confusion?

Answer 1

This is the unit circle I use to teach this topic. And specifically, to explain your issue with reference angles. Reference angle is simply the acute angle closest to the X axis. Therefore, 30, 150, 210, and 330 degrees all have 30 as their reference angle. Now, if you look at the (X,Y) values, comparable to the second graph you offered, only filled in for the common angles, you'll see that the absolute values of these 4 pairs of values are the same. More important, you can see that the sine value is the same for reference angle in quadrant I and II, and again for III and IV. Cosine is the same for quadrant I and IV, and then for II and III.

It's not until specific problems are offered that the importance of this concept comes to light. The student is given some complex problem which simplifies to $sin(x) = .5$ He looks at his calculator, which offers 30 degrees. It's by understanding the reference angle that he'll also get the second answer, 150 degrees.

The other way I describe all this - "Calculate the value of $cos(x)$ at 150 degrees. By hand." The points are still on the circle, but the values are removed. The student drops a line from the point on the circle, and forms a triangle with the base angle of 30. But I asked for the cosine of 150. Exactly. To get the values needed, the triangle itself has an angle of 30, and a base with a negative length.

In the classroom, the introduction and explanation of reference angles should have a flow from the discussion of all first quadrant observations. For the students who are attentive, there's no leap in understanding, just a continuation of learning.