When a student first encounters trigonometric radians and radian measure for angles measure, a teacher often sees him throw up mental blocks. The student looks at the material as something foreign because it is unfamiliar territory. One of the first concepts to conquer is changing from degree measure to radians or vice versa. The following is a process that I've found beneficial for easy and quick conversion of special angles that are multiples of 45 degrees or 30 degrees.

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After using the standard introduction to trigonometric radians and moving on to equate 180 degrees with pi radians, I demonstrate the normal process of multiplying by 180/pi or pi/180 to change forms. I do several examples where we change from degrees to radians and then have the students practice a few. I repeat that process with radians to degrees. However, in my list of about 10 examples for the students to try, I include problems such as 135, 180, 225, 210, and 300 degrees. After each set of examples, the work is checked and questions are answered with a demonstration of the process if needed. If more practice is needed, we try again.

When I feel that the students are comfortable with the process, I ask the class if they'd be interested in a quick and accurate method. Of course, the response is yes. I do explain that the method does not work for every angle -- it's only for "special angles." At this time, I have the students change 0, 30, 45, 60, and 90 degrees to radians, if they've not already done so in the previous examples.
I point out interesting facts such as 0 degrees is 0 radians and 30 degrees is pi/6 radians while 60 degrees is pi/3 radians. We spend a few seconds just memorizing the degree and radian equivalents.

To begin the demonstration of changing quickly, I use 150 degrees. I write the following for the students to see.

As I write, I comment that I'm simply rewriting 150 as 5 times 30 in the first step. In the 2nd step I'm substituting 30 degrees with pi/6 radians because we showed they were equivalent a few minutes back, and in the final step I'm simplifying the product. If this was one of the previously done examples, we check with the original solution.

Usually a student will ask something like, "How'd you do that?" or say "I don't get it." I then explain that I rewrite the original angle as a multiple of 30, 60, or 45 when possible and use the largest possible. I'll use an example such as 120 degrees to demonstrate. I'll write 120 degrees = 4(30 degrees) or 2(60 degrees) and say that since 120 degrees can be written as a multiple of 60 degrees, which is larger than 30 degrees, I use the 60 degrees method. I write the following for the students.

120 degrees = 2(60 degrees) = 2(pi/3 radians) = 2pi/3 radians. Again I explain my steps as I proceed. Keep in mind that for this article I'm using the words pi, degrees, and radians, but in my examples for the students I use the symbols for each. I then do a few more examples, being sure to include some that are multiples of 45 degrees and 60 degrees (135, 240, 225). My students next get more practice problems, we check, and answer questions.

Next I demonstrate changing from radians to degrees using the following process.
Explanation for each step is provided as I proceed.

-7pi/4 radians = -7(pi/4) = -7(45degrees) = -315 degrees

Again, then, practice problems are assigned and checked.

The closing of my demonstration and discussion involves significance of these "special angles." I tell the students that they should remember that any angle that is a multiple of 30 or 45 degrees is a special angle. I then ask what would be considered a "special angle" in radian measures; this is to emphasize that in radian measure the students should watch for multiples of pi/6 or pi/4. Sometimes it's necessary to discuss that if it's a multiple of 60 degrees or 90 degrees, it's automatically a multiple of 30 degrees and therefore "special." Often someone asks about changing 0 degrees or 180 degrees to radians. I tell them that 0 degrees (or 0 radians) is easy. 0 = 0! For 180 degrees, I remind that the largest angle in our list memorized earlier should be used. Since 180 degrees is a multiple of 90 degrees, I write

180 degrees = 2(90 degrees) = 2(pi/2 radians) = pi.

Then, I remind the class that we knew this from our discussion earlier in the class, but the example does show that our "shortcut" works.

Finally, I have the students randomly figure out and name those special angles between 0 and 360 degrees, inclusive. They, of course, are just finding numbers between 0 and 360 that are multiples of 30 or 45 degrees, but they're getting familiar with the process and learning the names of the special angles for recognition later. Those named are written for display and then rearranged in increasing order. If any are omitted, I simply say, "What about 210?" Each angle is then written in radian measure as well as degree so that we can look at the progression.

I have found this activity to help ease those anxieties about the dreaded trigonometric radians and to be an excellent preparation for the development of the unit circle and its applications. Good luck!

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