Trigonometry: Unit circle--do you require memorization? How to implement?

[cintinative]

Mr. Tarrou said he requires his students to memorize the radian values for the degrees. 

I am able to sketch and label it with the coordinates, and then use it for the problems. Is that how it is normally used? Or am I supposed to remember that pi/4 is a certain coordinate off the top of my head? Should I have my students memorize those coordinates? Or is it good enough that they have the values for 30-60-90 triangles and 45-45-90 triangles memorized so they can quickly write them in?

What tools have you used to help with this? Just a blank unit circle and fill it in?

Thanks in advance for your input.

Edited March 9, 2022 by cintinative

[cintinative]

@daijobu?

[daijobu]

One should memorize that a whole circle is radians.  After that you can derive everything else.  

At first you need to think very hard what a given radian corresponds to in degrees.  But then with experience solving lots of problems, they become automatic.  Eventually students will know the multiples of and the multiples of .   Learn how to calculate the equivalent of  negative degrees and degrees greater than .  I'll share an exam I took when I was in high school in another post.  

[daijobu]

Often I have difficulty inserting images into these posts.  I can see them and others do not, so LMK if you can't see them and I'll try to remember the workaround.  

 

Here's an early trig exam, where we were first introduced to radians and conversions.  Notice that we are only tested on multiples of 30 and 45 degrees.  

 

I include these other images to show that I am often sketching unit circles in the margins to remind myself what they correspond to.  

 

 

 

[regentrude]

7 hours ago, cintinative said:

Mr. Tarrou said he requires his students to memorize the radian values for the degrees. 

There is nothing to memorize about the radians to degree conversion. Students need to understand why 360 degrees is 2 pi. It is then completely obvious that 180 is pi, 90 is pi/2, and so on, for 30, 45, 60 and multiples.

Quote

 Or am I supposed to remember that pi/4 is a certain coordinate off the top of my head? Should I have my students memorize those coordinates? Or is it good enough that they have the values for 30-60-90 triangles and 45-45-90 triangles memorized so they can quickly write them in?

I am not sure what you mean by memorize "these coordinates". Are you talking about the values of sine/cosine? Or still about the angle?
All the student needs to understand is that the complete circle corresponds to 2 pi. He will then immediately know that pi/4 is an 8th of that, and thus has to be 45 degrees.
What do you mean by "values for the 30-60-90 triangles"? Are you talking about the values of sine and cosine functions? Yes, those should be memorized, which comes automatically through use in practice problems, not through rote memorization.
Or are you still talking about angles in radians? Those would simply be fractions of the complete circle in rad/deg.

Edited March 10, 2022 by regentrude

[cintinative]

8 hours ago, regentrude said:

There is nothing to memorize about the radians to degree conversion. Students need to understand why 360 degrees is 2 pi. It is then completely obvious that 180 is pi, 90 is pi/2, and so on, for 30, 45, 60 and multiples.

I am not sure what you mean by memorize "these coordinates". Are you talking about the values of sine/cosine? Or still about the angle?
All the student needs to understand is that the complete circle corresponds to 2 pi. He will then immediately know that pi/4 is an 8th of that, and thus has to be 45 degrees.
What do you mean by "values for the 30-60-90 triangles"? Are you talking about the values of sine and cosine functions? Yes, those should be memorized, which comes automatically through use in practice problems, not through rote memorization.
Or are you still talking about angles in radians? Those would simply be fractions of the complete circle in rad/deg.

I think that what he (Mr. Tarrou) wanted is that you could just say a 30 degree angle is pi/6 without spending the 10 seconds thinking about the fact that 180 degrees is pi and doing the division. He wanted quick recall. 

The coordinates are the (cos, sin) values for each radian/angle measure, yes. And yes, the values for the 30-60-90 triangles.  I wish I could tell you those had come to me through practice problems, but I still have to look up the 30-60-90 values on occasion. I can remember the 45-45-90 ones. LOL>

Edited March 10, 2022 by cintinative

[cintinative]

@daijobu the images didn't show up, unfortunately.

I got the impression that Mr. Tarrou makes his students fill out the unit circle often as a way of getting the values down and there in their minds for quick retrieval. I guess I am just trying to assess the best way to do this--use a blank unit circle and practice filling it in regularly?

The other day when I was doing some problems I sketched the unit circle so that I could use it for the problems. That helped me when I was finding coterminal angles, coordinates, etc. But I would think sketching the unit circle first would take a few minutes, so I was wondering if a normal math teacher would just expect you to know the values without sketching it. I guess that is sort of my question? 

Edited March 10, 2022 by cintinative

[regentrude]

3 minutes ago, cintinative said:

I think that what he (Mr. Tarrou) wanted is that you could just say a 30 degree angle is pi/6 without spending the 10 seconds thinking about the fact that 180 degrees is pi. He wanted quick recall. 

The coordinates are the (cos, sin) values for each radian/angle measure, yes. And yes, the values for the 30-60-90 triangles.  I wish I could tell you those had come to me through practice problems, but I still have to look up the 30-60-90 values on occasion. 

It may help to remember that in the 30-60-90 triangle,  the short side is half as long as the hypotenuse. So you immediately have the 1/2 for the sin 30 or cos 60.

the quick recall comes from using. I see no value in isolated memorizing - you always have the 10 seconds to divide  pi by whatever number.

[daijobu]

16 hours ago, cintinative said:

@daijobu the images didn't show up, unfortunately.

 

Argh.  This is perennial problem for me on these boards.   I will try again:  

My point with these images is when I was first learning the unit circle, I would draw them in the margins, sometimes with (x,y) coordinates and eventually just little tick marks.  It only takes a few seconds and I was doing this during a timed exam.  Then eventually I could just picture them in my head.  

I think it wouldn't hurt to have your student regularly complete a blank unit circle until s/he can be fairly automatic.  I never did that because I was using it in my homework.  

 

[regentrude]

I was able to see the images in your first post, @daijobu.  When you had just posted them. However  NOW they don't show anymore 

Edited March 11, 2022 by regentrude

[UHP]

On 3/9/2022 at 10:35 PM, regentrude said:

There is nothing to memorize about the radians to degree conversion. Students need to understand why 360 degrees is 2 pi. It is then completely obvious that 180 is pi, 90 is pi/2, and so on, for 30, 45, 60 and multiples.

On 3/10/2022 at 7:24 AM, regentrude said:

the quick recall comes from using. I see no value in isolated memorizing - you always have the 10 seconds to divide  pi by whatever number.

I think there is a memorization aspect to it. "360 degrees in 2 pi radians" is similar to "12 inches in a foot." Neither can be derived by pure thought so they need to be remembered.

Treating the conversions for common angles, 30 : 2pi/12, etc, as though they were vocabulary words to be memorized does seem wrongheaded. But for beginners it might be more than a ten-second task to figure the radians in 30 degrees: they have to set up and solve 

360/30 = 2pi/x

In a classroom setting, watching the calendar and the clock, maybe the teacher thinks he has a reason to ask the kid to memorize.

I just finished introducing angles to my 7-year-old (ignorant of pi), and the starting exercises were memorization: there are 360 degrees in a whole circle, there are 180 degrees in half a circle, there are 90 degrees in the corner of a square. I thought it was really effective.

 

[regentrude]

1 hour ago, UHP said:

I think there is a memorization aspect to it. "360 degrees in 2 pi radians" is similar to "12 inches in a foot." Neither can be derived by pure thought so they need to be remembered

No. 12 inches in a foot is completely arbitrary. The radians measurement is not. Once you understand that the ratio of arc length to radius for a slice of circle is only dependent on the angle and not on the size of the circle, and that THIS is what radians means, the equivalence of 2 pi being the full circle is obvious.

And without this understanding, the memorization of radians values as vocab words serves no real purpose.

Edited March 11, 2022 by regentrude

[UHP]

1 hour ago, regentrude said:

No. 12 inches in a foot is completely arbitrary. The radians measurement is not. Once you understand that the ratio of arc length to radius for a slice of circle is only dependent on the angle and not on the size of the circle, and that THIS is what radians means, the equivalence of 2 pi being the full circle is obvious.

Perhaps there is something very natural about 2 pi — though that factor of "2" could have come out another way, if we had to start from civilization from scratch. But 360 degrees in a circle is arbitrary, and requires remembering.

Using this number to figure the radians in 60 degrees — or understanding the relation of the angle to "the ratio of arc length to radius" —  requires some facility with fractions and ratios. I suspect many kids start learning trigonometry with a very shaky grasp on ratios.

[regentrude]

21 minutes ago, UHP said:

Using this number to figure the radians in 60 degrees — or understanding the relation of the angle to "the ratio of arc length to radius" —  requires some facility with fractions and ratios. I suspect many kids start learning trigonometry with a very shaky grasp on ratios.

That is an unfortunate truth, and these students don't stand a chance of actually understanding trigonometry. 
They would be much better served by solidifying their pre-algebra and algebra skills first.