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All of the topics I've posted are not that interesting and aren't useful in any way; Is there a way I can delete them?
Also, if you make the hypotenuse have length √2, something interesting will happen to the graphs of these functions (you can do this by multiplying a and b by √2).
So, I was browsing Wikipedia and came across the Right Triangle article's Altitude section.
Then I noticed that the altitude line splits the right triangle into two other right triangles.
Then I thought, "I wonder what would happen if I calculated the trig functions for one of these little triangles…"
So I came up with these steps:
[list=*]
[*]Take a triangle ΔPAB where ∠A=90°, ∠P=θ, and line PB has length 1; then find the altitude and call it line f.[/*]
[*]Mark the point where the altitude meets the hypotenuse as point F.[/*]
[*]Take the new triangle ΔPAF and name line PA line b as it is the base of the triangle, and name line PF line e.[/*]
[*]Take the other new triangle ΔBAF and name line BA line a, and name the line BF line d.[/*]
[/list]
After those steps have been done, calculate the trigonometric functions for ΔPAF at ∠P.
You should get something that looks like this: https://www.desmos.com/calculator/j5wuwvtvk6 (the last three variables are capitalized because I had to capitalize E and I wanted it to be consistent).
For now I'm going to call these:
Wait, did I just rediscover eliptic geometry?
I have come up with a geometry that is similar to spherical geometry. I defined it as being like Euclidean geometry except for that parallel lines must also intersect, which makes all straight lines except for perpendicular lines intersect in at least two places, perpendicular lines intersect an unknown amount of times. The only differences from spherical geometry I know for sure are: [list=*]
[*]The plane is infinitely extending and not on a sphere.[/*]
[*]Antipodal points don't exist.[/*]
[*]Lines have at least one point at infinity.[/*]
[*]The natural units of length and area are fixed, but currently unknown; it is unknown if a natural unit of angle measurement exists.[/*]
[*]Polar lines don't exist.[/*]
[*]The area of a triangle is unbounded.[/*]
[/list]A representation I came up with is where lines seem to bend in parabolic shapes away from some chosen reference point, with the bending getting more extreme the farther the line is from the point; lines that go through the reference point appear straight, and how extremely a line bends away from the point is defined by the natural unit of length. There are problems with this representation though: [list=*]
[*]If you have two distinct lines, there will be two points at which they will intersect, but if you move the reference point the intersection points will also move.[/*]
[*]A pair of perpendicular lines with at least one of the lines on the reference point intersect in one place while perpendicular lines off the reference point intersect in two places.[/*]
[*]Line segments move around in unknown ways when the reference point moves.[/*]
[*]The angle of intersection points might change with the position of the reference point but I don't know for sure.[/*]
[/list]
That was as easy as
Βεν Γ. Κυθισ wrote:Do you mean rather than ?it should be (a^m)n^a.
DANG IT! I forgot to move the variables around! And I forgot to confirm it with a calculator or my miiiiind.
If 1500 tickets
I think the answer might be 1500 tickets because if 1500 tickets, then 1500 tickets!
I learnt that sine, cosine, and hyperbolic tangent all have attractive fixed points.
And to make sure your understanding is right, you should realize why Cantor's diagonal proof for the reals does not work for the rationals (which can have infinitely many digits as well).
Another good question: Is there a set which is larger than the integers (can't be paired with them), but smaller than the reals?
There is no set with a cardinality greater than
Hi Βεν,
Nice contribution! Yes, the repeated iteration of the cosine function converges: actually, it converges to the fixed point of the cosine function,
That makes so much sense! I also just tried
and it converged to its fixed point, 0. Tan has a fixed point but it is not attractive; sinh becomes a vertical line on 0 which means there is no attractive point, 0; cosh has no fixed point but it converges to ∞; 0 is tanh's attractive fixed point; cot has no attractive point due to it being tan with the x axis shifted; sec is related to tan because it shares half of its discontinuous points with tan suggesting it has no attractive point, and it does have no attractive point; csc is sec with the x axis shifted so it has no fixed point.I found this website by looking up help on how to solve math problems, of course,
I found this site in a similar way! I wanted to know what the trigonometric functions were, specifically sine, cosine, and tangent. And the Math is Fun page was one of the top results that wasn't a Wikipedia article that expects you to know everything else in the field.
My first name is Benjamin, and if you are able to read ancient Greek, you could find out my last name. I like to do stuff sometimes. I like to playz the gamez (who doesn't?). I like to program; I like to use Python, Lua, and Scratch to code my stuff. I like to think of stories that almost always connect somehow to this huge complex megastory that I haven't finished yet (and no, I haven't uploaded any of my stories yet). I like to play around with physics simulators like The Powder Toy and Algodoo. I think about math a lot (which explains my presence on the Math is Fun fourm). I also go outside sometimes.
What does this mean?
So if you have f_0 = cos(n) and then f_1 = cos(f_0) and so on to get f_x = cos(f_(x-1)), and then let n be any real number, f_x converges to approximately
If you know anything that might relate to this new irrational number please respond to this post!Edit:
I found out what this is, so don't clog the post with what you think it means, its meaning has already been revealed.
Hi Βεν,
Welcome to the forum! Thanks for your contribution. That looks like a nice list. Some comments:
I know what you mean, but this can be misleading: is multiplied by itself times. For example, , where gets multiplied by itself one time.a^n=a multiplied by itself n times
The brackets should go around the here, i.e. for even .If n is even then (-a^n)=a^n
On one side of the equation, the and are the wrong way round. It should read:a^n÷a^m=a^(m-n) (Makes sense right?)
If n>0 then (a^m)na=a^(m+n)
What did you mean here?
I fixed my errors and typos. My answer to
What did you mean here?
is that I messed up; it shouldn't be (a^m)na, it should be (a^m)n^a. I feel so stupid for not checking with a calculator.
I have a piece of paper that has good exponentiation equations to memorize.
This is what is written on it at the time of this post (edited ones have an underline).
[list=*]
[*]a^n=a multiplied by itself n times n-1 times[/*]
[*]If n is even then (-a)^n=a^n[/*]
[*](abc⋯)^n=a^n×b^n×c^n×⋯[/*]
[*](1÷a)^n=1÷a^n[/*]
[*](a+b)^2=a^2+2ab+b^2[/*]
[*]a^m×a^n×⋯=a^(m+n+⋯)[/*]
[*]a^n÷a^m=a^(n-m) (Makes sense right?)[/*]
[*]((a^m)^n)^⋯=a^(mn⋯)[/*]
[*]If n>0 then (a^m)a^n=a^(m+n)[/*]
[*]2^n-2^(n-1)=(2^n)÷2[/*]
[/list]
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