Why is the Unit Circle Important in Teaching Trigonometry?

[summer of glr]

Ok- I am teaching trigonometry to low level students right now and I am trying to figure out why they need to know the unit circle. Are there some interesting things they can learn about by using a unit circle?

So far, we pretended it was a magic-barbie-sized-half-underground-ferris-wheel, which allowed them to generate sine and cosine wave graphs... but really, i have nothing interesting beyond that.

Please send help.

 

[AlephZero]

Pythagoras proved that sin^2 x + cos^2 x = 1.

(Well of course he didn't, because the ancient greeks never invented trig, but I hope you understand the joke).

 

[dkotschessaa]

I don't know how "new" this approach to learning trig is, but I know that just 13 years ago (is that long?) when I learned trig the first time, this is not the approach we used. I don't think it's that the unit circle is "new" so much that it seems newer in being used as a teaching device.

So I am in a pre-calc class now and our book is entitled "Precalculus: A unit circle approach."

If I knew what approach I used the first time I could tell you why this was better. But basically it allows you to not just learn but understand the trig functions of special angles like 30, 45, 60, and 90 degrees, and also the angle that correspond to them (sorry my language is not more precise) in other quadrants, such as 150, 135, and 120.

From my experience, what could amount to a lot of memorization (some people do try to *memorize* the unit circle, which I think is silly) boils down to understanding how the unit circle works.

Example: Find the exact value of the following function: cos(60 degrees)

The unit circle definition of the cosine function is the x coordinate over r (the radius, which is always 1).

The x coordinate at 60 degrees is 1/2. Therefore the cosine of 60 is .5

Other trig functions can be gotten similarly.

Once you learn other identities like sum and difference formulas, you can calculate a lot of other angles and get exact values (not the decimal approximations the calculator will give you).

For example if you need the cosine of 105, this is actually the cosine of 60 + 45. Once you know the formulas you can get an exact value for this angle.

Here is the wiki image for the unit circle:

http://en.wikipedia.org/wiki/File:Unit_circle_angles_color.svg

If you start to look at the patterns and understand *why* the values are what they are, it becomes very easy to commit to memory.

Really all you need is the first quadrant, 90, 180, 270 degrees and you can use reference angles to do other trig functions. I can explain more on that if you want.

Here you can find the Khan Academy videos that talk about the unit circle and it's use:
http://www.khanacademy.org/#Trigonometry


-DaveKA

 

[Unrest]

I also teach trig to higher level students and we use the unit circle. I think it can do two things:

1) Help them visualize why the trig functions are what they are. Not just arbitrary wavy lines.

2) Find exact values for certain ratios. Our tests require them to give some exact value answers.

3) Stop them having to memorize numbers, as dkot said.

4) Remind them that you can plot a circle using sin and cos instead of the ugly x^2+y^2=r^2. I'd have loved to know this when I was a student and trying to do computer graphics. But I had to pick it up from friends instead of the teacher.

I probably wouldn't use it with kids that would get confused and scared off by the concept and just want to memorize things to pass the test. But it seems like you have high expectations :P

 

[Goongyae]

Do you really have to teach using degrees of angle?

Radians are much more natural mathematically. For example:

1. The length of an arc of a unit circle is the same as the angle it forms.
2. sin(x) = x - x^3/3! + x^5/5! + etc.
3. sin(x) = (e^(ix) - e^(-ix))/(2i)
4. d^8/dx^8 sin x = sin x
5. lim (x->0)[ sin x / x] = 1

Using degrees:

1. The length of an arc of a unit circle is the same as the angle it forms, but multiplied by pi/180.
2. sin(x) = pi * x/180 - pi^3 x^3 /34992000 + pi^5 x^5 / 22674816000000 - etc.
3. sin(x) = (e^(i pi x/180) - e^(-ipix/180))/(2i)
4. d^8/dx^8 sinx = pi^8sin x/1101996057600000000
5. lim (x->0)[ sin x / x] = 0.0174532...

Why not throw in minutes and seconds of arc, and grads while you're at it. Also use roman numerals. That will really speed up their learning.

Degrees are clearly a tool of the Devil. For proof one need only consider the sine of the Beast: sin(666) = cos(6 x 6 x 6) = cos( 6^6^6 ).

 

[dkotschessaa]

Do you really have to teach using degrees of angle?

Radians are much more natural mathematically.

And yet ironically,they are significantly less intuitive and familiar for people starting out learning trig.

For example:

1. The length of an arc of a unit circle is the same as the angle it forms.
2. sin(x) = x - x^3/3! + x^5/5! + etc.
3. sin(x) = (e^(ix) - e^(-ix))/(2i)
4. d^8/dx^8 sin x = sin x
5. lim (x->0)[ sin x / x] = 1

Using degrees:

1. The length of an arc of a unit circle is the same as the angle it forms, but multiplied by pi/180.
2. sin(x) = pi * x/180 - pi^3 x^3 /34992000 + pi^5 x^5 / 22674816000000 - etc.
3. sin(x) = (e^(i pi x/180) - e^(-ipix/180))/(2i)
4. d^8/dx^8 sinx = pi^8sin x/1101996057600000000
5. lim (x->0)[ sin x / x] = 0.0174532...

Why not throw in minutes and seconds of arc, and grads while you're at it. Also use roman numerals. That will really speed up their learning.

Degrees are clearly a tool of the Devil. For proof one need only consider the sine of the Beast: sin(666) = cos(6 x 6 x 6) = cos( 6^6^6 ).

Well since you put it that way...

-M

 

[Landau]

Ok- I am teaching trigonometry to low level students right now and I am trying to figure out why they need to know the unit circle. Are there some interesting things they can learn about by using a unit circle?

How are you going to define the sine and cosine without the unit circle? What is the sine of 100 degrees?