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XM8 · Mar 21, 2009, 03:55 PM

Hi,

I've got a test coming up next Thursday about Unit Circles and I really need to ace it or I'm going to fail.

I'm trying my best to understand the whole chapter and have understood most of it but I just can't get around a few things.

On the unt circle, there are all sorts of values such are 3pi/4 ; 5pi/4 ; 11pi/6 ; 7pi/4 etc. etc.

I know that pi = 180 degrees, pi/2 = 90 deg ; pi/3 = 60 deg ; pi/4 = 45 deg and that pi/6 = 30 deg.

But what about the rest of the values? 3pi/2 must be 180 x 3 / 2 = 270 degrees. I understand that but isn't there some way to easily place the values on the circle?

My maths teacher showed me a calculation that goes like this. Say I wanted to find 7pi/6.

I'd do : 8pi/6 - pi = 7pi/6 . Ok well that's logical - but where do I put 7pi/6 on the circle?

I'd really appreciate any help.

Thanks,

-Xm8

galactus · Mar 21, 2009, 05:22 PM

7Pi/6 would go between 6Pi/6 and 8Pi/6, wouldn't it?

In other words, it would be between ${\pi}$ and $\frac{4\pi}{3}$ . Since $\frac{7\pi}{6}=210 \;\ degrees$

${\pi}=180 \;\ degrees$ and $\frac{4\pi}{3}=240 \;\ degrees$

To convert from radians to degrees, multiply by $\frac{180}{\pi}$

i.e. $\frac{\pi}{6}\cdot\frac{180}{\pi}=30$

After all, it's just a fraction.

Those values on the circle come from the sine and cosine of the various angles.

Take $(\frac{\sqrt{3}}{2},\frac{1}{2})$ on the circle at 30 degrees or Pi/6 radians.

$cos(\frac{\pi}{6})=\frac{\sqrt{3}}{2}$ and

$sin(\frac{\pi}{6})=\frac{1}{2}$

$\frac{\pi}{6}=30 \;\ degrees$

Since a unit circle has a radius of 1, we can use Pythagoras to find r.

$\sqrt{(\frac{\sqrt{3}}{2})^{2}+(\frac{1}{2})^{2}}= 1$

Does that help?

sarnian · Mar 22, 2009, 05:46 AM

What is Pi?? Pi is just a factor : the factor between the circle radius and circle perimeter.

Have a look here on this Wikipedia graphic !

In the graphic the diameter = 1, so the radius = 1/2, and Pi = 3.14
While you 'un-rolled' the circle you also turned 360°.
Pi D = 2 Pi R = 360° so 1 Pi R = 180°
So all these Pi values in your question represent parts of the 'unrolled' circle.
Just solve the amount of Pi and multiply that with 180°

Success with your test.

XM8 · Mar 23, 2009, 12:55 PM

Thanks to both of you for your helpful answers.

I've now understood the principle although there are a few things that are confusing.

In my text book (which by the way is not so helpful) there is an exercise where I have to place values on the unit circle.

Using the method given above it wasn't so difficult.

When I was given a value e.g. 5pi/4 all I had to do was multiply 180 with 5 and divide by 4 - easy.

However I did come across a few values which aren't really that easy.

For instance, 81pi/4. Now multiply 180 with 81 and you'll get 14,580. Divide that by 4 and you'll end up with 3645 degrees. Pretty crazy isn't it?

My maths teacher gave us the following method, however I hope someone can help me to understand it better...

He did 80pi/4 + pi/4 = 81pi/4

He then divided 80pi/4 and he got 20pi.

Of course this is all logical but there are a few things I don't understand.

Why did he chose to subtract 1 from 81? Why not add 1 to 80? Is there a difference?

And how come he placed 81pi/4 where pi/4 is? Is there supposed to be some connection here? Because if there is it would make sense, as he placed 71pi/3 and 97pi/3 where pi/3 is.

Sorry for all this, it's just that I really need to ace that test or I'm going to end up repeating the whole year with kids 2 years younger than me and I can't afford that.

Thanks,

-Xm8

sarnian · Mar 23, 2009, 05:43 PM

Hello XM8

As you know 2 Pi R represents a full circle. But 360° is the same in degrees as 0°.

81 Pi/4 = (80 + 1) Pi/4 = 80/4 Pi + 1/4 Pi = 20 Pi + 1/4 Pi = 10 full circles + 1/4 Pi (= 1/4 x 180° = 45°)

71 Pi/3 = (72 - 1) Pi/3 = 72/3 Pi - 1/3 Pi = 24 Pi - 1/3 Pi = 12 full circles - 1/3 Pi (= -1/3 x 180° = -60°)

97 Pi/3 = (96 + 1) Pi/3 = 96/3 Pi + 1/3 Pi = 32 Pi + 1/3 Pi = 16 full circles + 1/3 Pi (= 1/3 x 180° = 60°)

XM8 · Mar 24, 2009, 03:06 AM

Thanks for the answer.

I'm at school right now and a bit tired. I'll look at that when I get home and post back.

Thanks

-Xm8

XM8 · Mar 25, 2009, 11:12 PM

Hello Sarnian,

Thanks for the answer, but I'm still a bit lost here.

I understood the idea of how to get the angle I want to place, but my biggest problem is WHERE do I place it?

I was doing some exercises yesterday and I got 3pi/4. That makes 135 angles but I have no clue where to put it.

I realised that it was symmetrical to pi/4 by the y axis, but I don't know if that's just a coincidence...

During my maths test there will be one part where we're not allowed to user a calculator and I need to be fast.

Any advice?

Thanks,

-Xm8

Unknown008 · Mar 26, 2009, 02:04 AM

You start from your x-axis, going anticlockwise.

For angles such as $\frac{3\pi}{4}, \frac{\pi}{2}, \frac{\pi}{8}$ you can draw them easily enough. $\frac{3\pi}{4}$ is obatined by drawing a line dividing 90 degrees by two, $\frac{\pi}{8}$ is further diving that same angle by two, if you undesrtand what I mean.

Look at Galactus' posted attachment. You'll see your $\frac{3\pi}{4}$ at coordinates $(-\frac{\sqrt2}{2} , \frac{\sqrt2}{2})$ , and it is midway between $\frac{\pi}{2}$ and $\pi$ .

sarnian · Mar 26, 2009, 02:19 AM

Originally Posted by XM8

That makes 135 angles but I have no clue where to put it.

Draw the circle with x and y lines.
Right on x line = 0°.
Top y line = 90°.
Left x line = 180°.
Bottom y line = 270°.

135° = 90° + 45°
Draw line from 180° to 90° and connect.
Halve this line.
Draw line from center of circle via half point of that line to circle perimeter.
Where this line crosses the circle you have 135°.

In general : just draw a triangle based on the angle you need (use sin or cos).