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Geometry Question

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Geometry Question

Unread postAuthor: boyntonstu » Wed Oct 27, 2010 7:00 am

Here is a very interesting geometry question that I ask my students.

Using a piece of paper, a pencil, and a ruler, draw a 4 degree angle.
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Unread postAuthor: inonickname » Wed Oct 27, 2010 7:27 am

Draw a right angled triangle with the adjacent ~14.3 times longer than the opposite..the angle should be 4 degrees?

Eg. a 14.3 cm side, with a 1 cm side at 90 degrees to it..with a line between the points making up a right angled triangle. Should be close enough to four degrees
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Unread postAuthor: boyntonstu » Wed Oct 27, 2010 9:30 am

inonickname wrote:Draw a right angled triangle with the adjacent ~14.3 times longer than the opposite..the angle should be 4 degrees?

Eg. a 14.3 cm side, with a 1 cm side at 90 degrees to it..with a line between the points making up a right angled triangle. Should be close enough to four degrees



Sounds right.

Where did you get the 14.3:1 ratio?

Can you derive this ratio without using a tangent table by using one geometry rule?
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Unread postAuthor: Technician1002 » Wed Oct 27, 2010 10:05 am

Does this have anything to do with spudding? The last math thread was locked.
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Unread postAuthor: lozz08 » Wed Oct 27, 2010 10:08 am

sin(4 deg.)=(opposite side)/(hypotenuse)

sin of four degrees pretty much needs to be done with a calculator...

0.0697564737=(opposite side)/(hypotenuse)

we want to draw a simple triangle so lets make the opposite side 1cm:

1/0.0697564737=hypotenuse
hypotenuse=14.34
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Unread postAuthor: boyntonstu » Wed Oct 27, 2010 10:19 am

lozz08 wrote:sin(4 deg.)=(opposite side)/(hypotenuse)

sin of four degrees pretty much needs to be done with a calculator...

0.0697564737=(opposite side)/(hypotenuse)

we want to draw a simple triangle so lets make the opposite side 1cm:

1/0.0697564737=hypotenuse
hypotenuse=14.34


No trig tables may be used.

There is a much easier way.

In spudding is there ever a need to figure angles?
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Unread postAuthor: Ragnarok » Wed Oct 27, 2010 10:29 am

boyntonstu wrote:In spudding is there ever a need to figure angles?

Yes.
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Unread postAuthor: boyntonstu » Wed Oct 27, 2010 10:46 am

Ragnarok wrote:
boyntonstu wrote:In spudding is there ever a need to figure angles?

Yes.


Can you make a 4 degree angle with paper, pencil, and ruler without using trig tables?
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Unread postAuthor: Ragnarok » Wed Oct 27, 2010 11:18 am

Yes.

1) For an angle that small, then tan (x) will be close to X, if using radians. A 1 to (4/180*pi) ratio will give an angle of 4.003 degrees.

2) If you want a better approximation for larger angles, it's strictly sin(x) which is close to x when X is small, so using the cos(x)<sup>2</sup> + sin(x)<sup>2</sup> = 1 rule, you can get a better approximation of tan(x) that's 95% accurate up to about half a radian.

3) If you're really in a silly mood, you can derive tan(x) by solving the sequences
sin(x) = x - x<sup>3</sup>/3! + x<sup>5</sup>/5! - x<sup>7</sup>/7! + x<sup>9</sup>/9! - ......
cos(x) = 1 - x<sup>2</sup>/2! + x<sup>4</sup>/4! - x<sup>6</sup>/6! + x<sup>8</sup>/8! - ......


as far as you can be bothered (or one of them, and find the other via cos(x)<sup>2</sup> + sin(x)<sup>2</sup> = 1)
Then feed them into tan(x) = sin(x)/cos(x) and go from there.

However, they invented these wonderful things called protractors and calculators, so normally, I don't bother.
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Unread postAuthor: skyjive » Wed Oct 27, 2010 11:20 am

Is 4 degrees just pulled out of a hat or is it special? I suspect that there's a trick involved specific to that angle, and that you're not looking for a brute force solution using a Taylor series or anything like that.
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Unread postAuthor: boyntonstu » Wed Oct 27, 2010 11:24 am

Excellent!

The nice thing to remember is that 1 radian is 57.3 degrees.

Therefore 1 part in 57.3 is 1 degree.

Go out 57.3 units and up 1 and you have 1 degree.

Very simple to do.

I designed and built a truck wheel alignment business based on this relationship.

We aligned fire engines at the fire stations.

Next challenge: How to measure the camber angle of a tire/wheel to within 1/10 of a degree with a homemade $5 device?
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Unread postAuthor: Ragnarok » Wed Oct 27, 2010 11:38 am

boyntonstu wrote:How to measure the camber angle of a tire/wheel to within 1/10 of a degree with a homemade $5 device?

Seems optimistic. You might get a precision of 0.1 degrees, but I don't think you're likely to get an accuracy of 0.1 degrees.
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Unread postAuthor: boyntonstu » Wed Oct 27, 2010 11:42 am

Ragnarok wrote:
boyntonstu wrote:How to measure the camber angle of a tire/wheel to within 1/10 of a degree with a homemade $5 device?

Seems optimistic. You might get a precision of 0.1 degrees, but I don't think you're likely to get an accuracy of 0.1 degrees.


I can get accuracy of 0.1 degree.
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Unread postAuthor: MrCrowley » Wed Oct 27, 2010 3:24 pm

I will leave this topic open since it has received quite a bit of feedback, is so very slightly related to spudguns and so you don't post another topic like this because we locked your previous ones.

However, Boytonstu, please don't post another topic like this (math related, pointless, irrelevant). You can come up with as many examples of other topics that weren't locked as you want but at the end of the day the decision is down to the Moderator(s). Not every topic that should be locked will be locked, there are a few reasons for this, some of which you may be unaware of as the reasons could be discussed over PM.

This topic itself could end up being an example of a topic that wasn't, but should've been, locked. So, from now on, Boyntonstu, keep your NSGRD topics within the guidelines. If you purposely post a topic that falls through the cracks just to prove a point, well, we'll probably lock it just to prove we can. Usually I wouldn't go out of my way to say all this but seeing as we locked a math related topic of yours yesterday for being irrelevant and then today (NZ time), you post another math related topic, I feel you're just doing it to try and prove a point.
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Unread postAuthor: Moonbogg » Thu Oct 28, 2010 12:53 am

L enter
@10<4

There. 10" long, 4 deg angle. Oh wait, that wasn't with a ruler, paper and pencil.
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