Philosophers and rhetoricians, Welcome!

Could some one PLEASE explain perhaps with diagrams (I feel they may indeed be necessary) the reasons wherefore these values are. It has to do with pythagoras' theorem I know that and I also know that I should know this and there was a point a few weeks agone when steven showed me that I did know.

phpbb

Episcopus
Textkit Zealot

Posts: 2563
Joined: Sat Jun 14, 2003 8:57 pm

Episcopus wrote:Could some one PLEASE explain perhaps with diagrams (I feel they may indeed be necessary) the reasons wherefore these values are. It has to do with pythagoras' theorem I know that and I also know that I should know this and there was a point a few weeks agone when steven showed me that I did know.

http://en.wikipedia.org/wiki/Sine

Imagine that the angle A is actually a hinge, and the segment h is like a clock hand, it can rotate around. But not exactly like a clock hand, because the segment b is fixed in length, but that the segments a and h grow or shrink, because they are defined by where a and h intersect. So, if we slowly rotate h clockwise (make the angle A = [face=SPIonic]q[/face] smaller) then the length of h and the length of a slowly become smaller. The angle C stays fixed at 90 degrees.

In this diagram, where growing or shrinking [face=SPIonic]q[/face] causes h to rotate, sin([face=SPIonic]q[/face]) = a/h, cosine([face=SPIonic]q[/face]) = b/h, and tan([face=SPIonic]q[/face]) = a/b.

Remember that 1/sqrt(2) = sqrt(2)/2, so that we can say:

sin(30) = sqrt(1)/2
sin(45) = sqrt(2)/2
sin(60) = sqrt(3)/2

Perhaps it's more palatable that way.

Democritus
Textkit Fan

Posts: 331
Joined: Fri May 07, 2004 12:14 am
Location: California

Interesting, we just use calculators at school.
phpbb

Emma_85
Global Moderator

Posts: 1564
Joined: Thu Jul 03, 2003 8:01 pm
Location: London

Emma_85 wrote:Interesting, we just use calculators at school.

We're not allowed to use calculators in uni maths exams.
phpbb

Eureka
Textkit Zealot

Posts: 741
Joined: Tue Feb 17, 2004 3:52 am
Location: Melbourne, Australia

We're not allowed to use calculators in uni maths exams.

ARgh....

I LOVE my calculator, I need it... it's my precious
phpbb

Emma_85
Global Moderator

Posts: 1564
Joined: Thu Jul 03, 2003 8:01 pm
Location: London

We are talking about PROPER HARDCORE SHUT THE **** UP AND DIFFERENTIATE mathematics exams here! WE DO NOT USE CALCULATORS WE CALCULATE SIN COS TAN SECT USING BUT A PEN PAPER COMPASS RULER. TRUE DAT. During said exams maths teachers slap us across the face with arbitrate geometrical instruments because it's SO BLOODY HARCORE!
Also at the end if we don't shrug 150kg we fail anyway. Because the traps must be a parabola.

We be hardcore. We take no calculators. We don't need it. We be knuckling down no pu\$sy footing around here lads. Calculators are for women and farmers as Emma has for us so well demonstrated.

*I also use jaffa cakes as shurikens during the exam, because it is allowed, because it is just HARDCORE round these parts. If you get knocked out and fail at your LIFE I was differentiating didn't throw anything sir. Get him he's integrating log x scum!
phpbb

Episcopus
Textkit Zealot

Posts: 2563
Joined: Sat Jun 14, 2003 8:57 pm

The explanation in the wikipedia is quite unconvenient, though.
You'd better draw a circle of diameter 1.
Draw a diameter through the center of the circle. and it meets the circle at A and D.
And from A draw a slant line AB that makes angle th with the diameter AD.
You get an arc(or arch) BD. And the line segment BD of this arch, or arc is the sine of th.
And the line AB is the co-sine of th.
So then what is a tangent? If you draw a perpendicular lineDC from D, and this line DC will be tanget to the circle, and extend AB until it meets the tangential line at C. Now the length DC, is the tangent of th.

http://blogfile.paran.com/BLOG_17207/20 ... 7_trig.gif

Now with this figure in mind, you can easily see that with angles pi/6, pi/4, pi/3, pi/2 you can draw regular triangle, square, hexagon, and just the diameter itself, respectively, on the circle with little difficulty. And you can figure out the values of sine, cosine, and tangets of the angles after the Pythagorean formula. I hope this helps. But I'm terribly bad at explaining.

I wanted to make this lengthy and detailed. But I was in a hurry 'cause I have to go home soon and help my wife.

edit:
This definition of the trigonometric functions might be unfamiliar. They are usually defined in terms of radius, rather than diameter. But I thought this is a philological forum and the etymologies fitted the better this way. And the definitions in accord to radius can be easily derived by drawing from the center of the circle a line parallel to AB and using the similarities. A trivial step.

edit again:
Why don't you start reading Elements in Greek?

mingshey
Textkit Zealot

Posts: 1318
Joined: Tue Aug 19, 2003 6:38 am
Location: Seoul