I know how to convert degree to radian vice versa. But how would you convert a real number to radian measure?
I was thinking about this when we were studying the wrapping function in class.
Consider the wrapping function. Point pi/3 lies on a real number line and the number line is wrapped around the unit circle. The point that pi/3 lands on on the circle is (cos,sin) or (1/2, (sqrt3)/2). Now how would you do 73? Maybe I was asking the wrong thing (that there is no conversion involved at all). Finding the cos and sin of pi/3 (or 60 degrees) is easy, but how would you find cos/sin of 73 without a calculator? Maybe I'm asking how to put 73 in terms of pi. BTW Jameson, how did you make "pi/180" look like that.? Hope that clears everything up.Originally Posted by MathGuru
Umm, now I get. Or do I?Originally Posted by c_323_h
Never heard about your wrapping function before but basing from your example here, it means the location of a point on the vertical numberline when plotted on the circumference of the unit circle by wrapping the length of the numberline on the circle.
Okay, let us see.
A unit circle has a radius of 1.
The length of the numberline up to the given height is equal to the circular arc on the unit circle.
>>>arc = (central angle)*(radius) ----------***
So, for your given pi/3 on the vertical numberline,
pi/3 = (central angle)*1
Hence, central angle = pi/3 also, but in radians now.
See, on the numberline or circumference, the given pi/3 is in real number, while the central angle of the unit circle for that given length is pi/3 radians.
Is that what you want to know, or is it related to that?
Your confusion came about because the circle is a unit circle, whose radius is always 1 or unity, whereby an arc and its corresponding central angle always have the same numerical value. Same numerical values but not same in units. [Zeez, are there no other term to use here other than "unit"? Using the "unit" itself to mean different meanings in the same paragraph is confusing itself.]
On lengths of arcs, the unit is in unit length. [Heck.]
On the central angles, the unit is in radians.
That is why the ancient ones were/are correct in using degrees for the unit of angles. Not radians.
[I think "radian" was invented by the Pure Mathematicians again to continue alienate/confuse us poor people when we delve into Math. Yeah.]
Let us call the central angle as angle C.
You want the location of the pi/3 of the numberline in the form (x,y).
x = (radius)*cosC = 1*cos(pi/3 rad) = 1/2 ----in unit length.
y = 1*sinC = sin(pi/3 rad) = sqrt(3) /2 --------in unit length
Unit length may be in inches, feet, meters, stones, ....
For the 73 on the numberline.
Arc = 73 ----after wrapping.
So, arc = central angle * radius
73 = C*1
C = 73 radians
x = cos(73 rad) = -0.7362 of a unit length
y = sin(73 rad) = -0.6768 of a unit legth
That means when wrapped on the unit circle, 73 unit lengths will map at (-0.7362,-0.6768). That is in the 3rd quadrant.
One complete circle or circumference for the unit circle is P = 2pi(1) = 2pi unit lengths
2pi = 2(3.1416) = 6.2836 unit lengths.
Now, 73/6.2836 = 11.6175 times.
Meaning, the 73 will wrap around the unit circle in 11 times plus 0.6175 of a time or 0.6175 of a perimeter/circumference.
0.6175(6.2836) = 3.88 unit lengths
The central angle for that is,
3.88 = C*1
C = 3.88 radians ------that is a tad more than pi or 3.1416 radians, so it is in the 3rd quadrant. --------------***
Then just use the formula arc = (central angle, in radians)*(radius).Originally Posted by c_323_h
Say, radius = 2 unit lengths.
73 = C*2
C = 73/2 = 36.5 radians
x = (2)cos(36.5 rad) = 0.7264
y = (2)sin(36.5 rad) = -1.8634
Therefore, on a circle whose radius is 2 unit length, the 73 unit lengths on the numberline will fall at (0.7264,-1.8634). That is in the 4th quadrant.
Circumference or perimeter of the 2-radius circle , P = 2pi(2) = 4pi = 12.5664 unit lengths.
73/12.5664 = 5.81 times.
That is wrapped 5 times and 0.81 of a time or circumference.
0.81*12.5664 = 10.1788 unit lengths.
arc = C*2
C = 10.1788/2 = 5.0894 radians
4th quadrant is from (1.5pi) to 2pi radians.
That is (1.5pi =) 4.7124 rad to (2pi =) 6.2832 rad.
C=5.0894 rad is in between those two limits, hence C is in the 4th quadrant.
Is there a way you could convert 73 in terms of pi? I think this in what I was trying to ask is the first place, but I was getting radians and real numbers confused. Here is what I tried to do:
73 = (73/2pi) x 2pi
= 36.5/pi x 2pi
and then find coterminal angles and reference angles to find cos, sin???
73pi/2 - 72pi/2 = pi/2
which is probably wrong because I think it's hightly unlikely to be at 90 degrees or (0,1). and i calculated to wrap 18 times where you calculated 11. where did my logic go wrong?
I see.Originally Posted by c_323_h
What is 73 unit lengths in terms of pi?
pi(x) = 73
x = 73/pi
That means 73 = (73/pi)(pi)
No sense, isn't it?
But if we use pi=3.1416 or the exact pi in the calculator 0n the 73/pi, we will get
73/pi = 73/3.141592654... = 23.23662169
Therefore, 73 -->>(23.23662169)pi.
But that is not in radians. That is still in unit lengths
You need to involve the circle where you have to wrap the 73 unit lengths.
On a unit circle.
Arc = (23.23662169)pi unit lengths.
Perimeter of unit circle = 2pi(1) = 2pi unit lengths.
Wrap that Arc onto the unit circle,
(23.23662169)pi / 2pi = 23.23662169/2 = 11.61831085 times.
Is that the same as was found before?
On a 2-unit length radius circle
Length of arc = (23.23662169)pi unit lengths
Perimeter of the circle = 2pi(r) = 2pi(2) = 4pi unit lengths.
Wrap that Arc onto the circle,
(23.23662169)pi / 4pi = 23.23662169/4 = 5.809155423 times.
Is that the same as was found before?
No problem. I just cannot stay all the time here. I read many websites---Steelers' included---among other things to do, so I cannot answer your questions at once.Originally Posted by c_323_h
Exact value? Of pi?
Umm, you are not one of them Pure Mathematicians, are you?
The exact value of pi depends on who wants to know it. Pure Mathematicians are still spending time, effort and money to come up to the next length of the decimal places of pi. Like the world will explode if they cannot continue finding the next last digit of pi. If that is what you want to find out, then search the Web re pi.
For computing purposes in your wrapping function, the exact pi lies inside your calculator. Press the pi key for pi. [Man, that was difficult to explain!]
Umm.Originally Posted by c_323_h
Let me refer you to the (1/2,sqrt(3)/2) of the location of the wrapping of pi/3 in your first example. Are they in terms of pi?
Those, and (-0.7362,-0.6768), are coordinates on the x,y axes. How or why should they be in terms of pi?