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(Message started by: tiber13 on Mar 14th, 2007, 11:55am)
Title: basic math terms
Post by tiber13 on Mar 14th, 2007, 11:55am
yes , I know i am a BIG noob for this but< what do sin and tan mean?
Title: Re: basic math terms
Post by towr on Mar 14th, 2007, 12:12pm

on 03/14/07 at 11:55:36, tiber13 wrote:
yes , I know i am a BIG noob for this but< what do sin and tan mean?

Maybe the easiest way to look at it is to draw a unit circle (i.e. it has a radius of 1 unit). Then draw two lines horizontally and vertically through the circle.
Now if you pick a point on the circle, and draw a line through that point and the center, then the sine of the angle between that line and the horizontal one is the (shortest) distance from the point the horizontal line (negative if it's below it).
And the cosine of that same angle is the shortest distance from the point to the vertical line (negative if it's to the left of it).
The tangent of the angle is the division of the sine by the cosine, and it's also the slope of the line through your point.

(And for completeness sake, you have to take the angle counter-clockwise starting at the horizontal line. That's important because two crossing lines have two different angles (each twice), and you need the right one)

I suppose a discussion of trigonometric functions really needs pictures. But there are some at http://en.wikipedia.org/wiki/Trigonometric_function
Title: Re: basic math terms
Post by Sir Col on Mar 15th, 2007, 12:41am
I've also got an FAQ that attempts to answer this on one of my websites:
http://mathschallenge.net/index.php?section=faq&ref=geometry/trigonometric_identities
Title: Re: basic math terms
Post by Grimbal on Mar 15th, 2007, 3:17am

on 03/14/07 at 11:55:36, tiber13 wrote:
yes , I know i am a BIG noob for this but< what do sin and tan mean?

Sin is something bad, don't do it.
Tan is what you get if you stay under the sun.
::)
Title: Re: basic math terms
Post by tiber13 on Mar 15th, 2007, 12:27pm
once again i need the "Eleven Year's Old Guide to William Wu's Puzzle Forum Dummies Guide Illistrated (that means it got pictures)cuz i didnt understand a word ??? ??? ??? ??? ??? ??? ??? ??? ??? ??? ??? ??? ???
Title: Re: basic math terms
Post by tiber13 on Mar 15th, 2007, 12:29pm
if you could reword it using up to 6 letter words i would get it. ^_^
Title: Re: basic math terms
Post by towr on Mar 15th, 2007, 1:51pm

on 03/15/07 at 12:29:34, tiber13 wrote:
if you could reword it using up to 6 letter words i would get it. ^_^

The tangent can't be explained using up to 6 letter words. If only because it is itself 7 letters. ::)

It really needs illustration; and I suppose preferably it should be taught in person. Don't you have teachers that can teach you these these things?
Title: Re: basic math terms
Post by Icarus on Mar 15th, 2007, 4:33pm
They usually don't teach that stuff over here until high school. If you don't know about algebra yet, you can just forget the following, as you will need to learn the basics of algebra before the rest of it has any hope of making sense.

There are 6 trigonometric functions, though we mainly use three of them:

sine      cosine      tangent

Take a right triangle, like the one in the picture below. Suppose I magnify it so that every side increases in length by the same proportion, I find that the angles stay the same size. Because the changes are proportional, the ratios of the sides to each other also do not change. This means that for a given angle, the ratios of the sides are unchanged, no matter how big the triangle is.

This is a very, very, very useful fact! Suppose I want to find out how tall a tree is. Trying to climb up into the tree to the highest point, find a clear spot I can drop a measuring tape straight down on making sure the top of my tape is level with the top of the tree (which won't have a clear path down at its location!), and reading off the length is a complicated and risky means of measuring the tree. On the other hand, I can spot that highest point from the ground fairly easily, if I am far enough away. I can also measure the angle from the horizontal ground up to the line-of-sight to the top of the tree fairly easily, as well as the distance from the place of my measurement to a point directly under the highest point of the tree (alright - sometimes you can't do this - there is a way around this problem, but to make things simple, we will just assume you can). That gives you a right triangle, like the one below, in which you know what the value of http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif is, and what the value of x is. But any right triangle with that same angle http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif in that same position is also going to have the same value for the ratio y/x. So go draw a small right triangle with the same angle http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif, measure its base and height, and find what (y/x) is for it. Multiply it by the x of the big triangle to get the height of the tree.

Since the ratio is going to be the same always, it makes sense to figure out what it will be ahead of time for any angle http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif, and just record this information in a table. Then, when you need to do this sort of measurement, you measure the angle, then look of the corresponding ratio in the table, rather than having to figure it out again. This is what people did for centuries - until the invention of calculators gave an easier method of calculating the ratios.

For each angle http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif, the ratio y/x is called the "tangent of http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif", and is denoted (written down) by "tan http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif". The ratio y/r is called the "sine of http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif" and is denoted by "sin http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif". The ratio x/r is called the "cosine of http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif" and is denoted by "cos http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif".

So, for example,
tan 45o = 1  because in a right triangle with a 45o angle, the two shorter sides (called the "legs") are the same length, so their ratio is just 1.

Some other well known values:
tan 0o = 0     sin 0o = 0     cos 0o = 1.
tan 30o = (http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif3)/3     sin 30o = 1/2     cos 30o = (http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif3)/2
tan 45o = 1     sin 45o = (http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif2)/2     cos 45o = (http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif2)/2
tan 60o = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif3     sin 60o = (http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif3)/2     cos 60o = 1/2
tan 90o = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif    sin 90o = 1     cos 90o = 0
Title: Re: basic math terms
Post by SWF on Mar 15th, 2007, 8:18pm
sin is short for "sine", and it is a function so is usually written with parentheses after it as in sin(x), where x can represent a number. sin(x) has a value which depends on x and is given by the formula:

  sin(x) = x - x^3/(1*2*3) + x^5/(1*2*3*4*5) - x^7/(1*2*3*4*5*6*7) + ...

The "..." means that the pattern in that formula continues on forever (which is why it is easier to abbreviate that complicated formula as sin(x) ). Whatever number you put in for x, that formula allows you to figure out what sin(x) is, but it turns out the value will never be more than +1 or less than -1.

The thing that makes sin(x) useful is that it relates to something moving around a circle at constant speed. If you have a clock on the wall whose center is 5 feet above the ground and the second hand is 1 foot long, the distance the tip of the second hand is above the floor changes with time. A formula for that distance involves sine, and that formula turns out to be:

(distance of tip of second hand from floor) = 5 + sin(x*0.10472)

For this formula to work, x represents the number of seconds since the second hand last passed the 9.
Title: Re: basic math terms
Post by Icarus on Mar 15th, 2007, 8:58pm
Actually, I would say that it is more often written without the parentheses than it is with them. Since some other threads show that tiber13 is already familiar with factorials, we can write the formula in this form:

sin(x) = x - x3/3! + x5/5! - x7/7! + x9/9! - ...

However, sine and cosine and tangent were known for about 1000 years before this formula and its cousins were discovered. Also, in order to use it, you have to measure angles in "Radians", not degrees. (Radians are a natural measure for angles. A full circle is 360 degrees, but only 2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif radians - about 6.28.)

cos(x) = 1 - x2/2! + x4/4! - ...

tan(x) also has a formula, but it is not nearly so nice. It is easier to calculate by tan(x) = sin(x)/cos(x).
Title: Re: basic math terms
Post by SWF on Mar 15th, 2007, 9:21pm

on 03/15/07 at 20:58:39, Icarus wrote:
Since some other threads show that tiber13 is already familiar with factorials,
I will be interested to see if he understands the theta that appears in your description.  Mathematicians love their Greek letters, in this thread (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1172436989) you use a Greek letter that isn't even in the statement of the question.
Title: Re: basic math terms
Post by SMQ on Mar 16th, 2007, 5:44am
Um, SWF, your "this thread" link is the URL for the "Modify" action on your own post...

:)

--SMQ
Title: Re: basic math terms
Post by Icarus on Mar 16th, 2007, 3:35pm
I considered that, but http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif is traditional, and it didn't seem to me that a roman letter would be much clearer.

And yes we do like our greek letters, and for good reason! If this mysterious thread is the one I am recalling, I chose to use http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/alpha.gif instead of a like the problem statement because when you use an a in the middle of a sentence, it is hard to tell quickly if you mean an a or a variable a or the word a. Since http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/alpha.gif is commonly used as a variable in that application, I chose to do this so that the appropriate interpretation would be instantly clear. I figured it likely that if the poster could have easily inserted http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/alpha.gif (thanks again, SMQ!), he would have done so, and only used a as a substitute.
Title: Re: basic math terms
Post by SWF on Mar 16th, 2007, 6:37pm
Oops, I took the link from the wrong tab in my browser. Maybe certain browser companies really did have my best interests in mind back when they were making excuses for not adding tabs in their browser. The link is fixed now, and Icarus is remembering the right one. True using "a" as a varable in line with text can be hard to read, but it isn't a problem if you write trig functions as "sin(a)" instead of "sin a" or "sina". Looking back on that thread Icarus did use parentheses with sine.  :-)


Quote:
However, sine and cosine and tangent were known for about 1000 years before this formula and its cousins were discovered.
Maybe being stuck on the triangle connection is why there was so little progress in mathematics in those 1000 years compared to the time after the formula was discovered.  :-)

I was just kidding, but there is some truth to what I was saying. I think the reasons sine appears so frequently in practical applications is more directly tied to it being part of the solution to certain linear differential equations rather than finding side lengths of triangles.
Title: Re: basic math terms
Post by Icarus on Mar 16th, 2007, 7:40pm
It is true the differential nature of trig functions is more intrinsic to their use than the triangle relations.

However, the triangle relations are considerably more elementary, and their application much easier for the layman to understand. Given the target audience for these posts, they seemed more appropriate.

However, it doesn't seem to be effective, since I know that tiber13 has visited today, yet he has made no post here. I fear that we have still lost him. :-/


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