Mathematics - 2018-01-18 (page 4 of 8)

Secret

2:09 PM

us.metamath.org/mpeuni/mmtheorems.html

Tuki

helo

Abhas Kumar Sinha

hi

does anyone is good knowing radians and degrees?

Tuki

@AbhasKumarSinha What's on your mind ?

Abhas Kumar Sinha

I'm unable to understand how sin pi/4 = cos pi/4

sine ratios are made to work with degrees not radians

If we'd consider radians in sin, cos. sec, cosec, tan and cot then those would not work

philmcole

@AbhasKumarSinha I think you are confused. The sine and cosine have nothing to do with degrees. Degrees is a unit. It's the same as if you say the parabola has to do with meters...

Abhas Kumar Sinha

2:24 PM

@Tuki @Secret @Semiclassical you know about that?

parabola?

what is that?

philmcole

$f(x)=x^2$

Abhas Kumar Sinha

No. I'm talking about trignometery

How sin pi/4 = cos pi/4

?

philmcole

Do you know what the values are of sine or cosine?

How did you define them?

Abhas Kumar Sinha

Also is there any value of radians = degree in magnitude? so that I can calculate pi

sin = perpendicular/hypotenuse

I mean sin A = side opp. to A / longest side of a triangle

philmcole

0

Q: how do i prove that $\sin(\pi/4)=\cos(\pi/4)$?

It's weird that i have not defined the tangent function yet. how do i prove that $\sin(\pi/4)=\cos(\pi/4)$? I have prove that $\tan:(-\pi/2,\pi/2)\rightarrow\mathbb{R}$ is a strictly increasing continuous bijection. (Not yet proved that it's a homeomorphism; i think i can show that arctan is co...

real-analysis pi

Abhas Kumar Sinha

2:28 PM

I tried to see those, but answers were complex and not of my level @philmcole

Secret

recall how radians is defined, the angle formed by a sector with arc length 1 and radius 1

philmcole

Then you should reply to them or say this in the question that you want a simple derivation @AbhasKumarSinha

Secret

meanwhile, degrees corresponds to 1/360 th of a circle

Abhas Kumar Sinha

0

Q: Simpler Derivation of $\sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}$,

As far I've studied the Basic Trigonometry in School, those are below - $$ \frac{1}{\sin \theta} = \csc \theta$$ $$\frac{1}{\cos \theta} = \sec \theta$$ $$\frac{1}{\tan \theta} = \cot \theta$$ And Angle Relations like - $$\sin \theta = cos(90 - \theta)$$ $$\tan \theta = \cot (90 - \theta)$$ $$...

trigonometry proof-writing circle triangle alternative-proof

yea, I understand i radians = 1/2 circumference / radius

philmcole

@AbhasKumarSinha You accepted an answer. Is it not good? Then you should reply to it and ask what you don't understand.

Abhas Kumar Sinha

2:32 PM

I replied but he didn't replied me back

I understood half of it

philmcole

So you've asked for example why $\pi = 180$. This is of course not true, as $\pi = 3,14159265... \neq 180$. What we say is that $\pi$ radians $= 180$ degrees.

Abhas Kumar Sinha

okay

philmcole

It's the same as if you say $60 = 1$ which is wrong. But indeed are 60 seconds $=$ 1 minute, right?

Abhas Kumar Sinha

okay

So, can I do this? $$/sin /frac{/pi}{4} = /sin 45$$

So, can I cancel sin both sides and say that /frac{/pi}{4} radians = 45 deg?

Semiclassical

1) \sin not /sin and same for \frac

Abhas Kumar Sinha

2:36 PM

wooops

Semiclassical

2) Yes to the second, no to the first

Abhas Kumar Sinha

..? didn't get that

@Semiclassical

Semiclassical

First would be sin(pi/4 radians)=sin(45 degrees)

Abhas Kumar Sinha

yep, so can I rewrite that as pi/4 rad = 45 deg

?

Semiclassical

It’s common to omit ‘radians’ on the left-hand side, but you never omit degrees

Abhas Kumar Sinha

2:38 PM

cancelling sin both sides?

Secret

you cannot cancel sin, it is a function, not a number

Semiclassical

I’d say it in reverse. 45 degrees are the same angle as pi/4 radians

Abhas Kumar Sinha

So, what is the difference between my statement and yours?

I can't cancel sin but I can write that?

Secret

The difference between radians and degrees hinge on how you define a smallest angle

Semiclassical

So taking the sine of 45 degrees gives the same result as sine of pi/4 radians

Secret

2:41 PM

Degrees define a unit of angle by dividing a circle into 360 sectors and the angle of each sector is treated as 1 degrees

Abhas Kumar Sinha

So, Is there a value of degrees = radians (any magnitude) so that, I can calculate pi theoritically

Lozansky

"Determine the coefficients $a_n$ such that the function $$u(r,\theta) = \sum_{-\infty}^{\infty} a_n r^{|n|} e^{i n \theta}, \: 0 \leq r \leq 1, \: 0 \leq \theta 2 \pi$$ satisfies $u(1,\theta) = \delta(\theta - \theta_0)$

Semiclassical

The reason I would avoid it is that, for instance, sin(0 degrees) = sin(180 degrees) = 0

Secret

Meanwhile radians is defined by a circle of radius 1 and arc legnth 1. We call the angle of the resulting sector 1 radians

Abhas Kumar Sinha

So, the radians are always smaller than magnitude when compared to degrees?

Semiclassical

2:42 PM

But you can’t ‘cancel’ sine from both sides in order to conclude that 0 degrees = 180 degrees

So I’d focus less on “they have the same sine” and more “they define the same angle”

Secret

So if you take 45 pieces of the sector that is used to define degrees, you get 45 degrees. But you need much less pieces of the sector that is used to define radians in order to construct a sector of the same area, because the sector corresponding to 1 radian is bigger than that of 1 degrees

Lozansky

So since $\delta(\theta - \theta_0) = \dfrac{1}{2\pi} \sum_{-\infty}^{\infty} e^{in(\theta - \theta_0)}$, it's obvious $a_n = \dfrac{e^{-in \theta_0}}{2\pi}$

Secret

So the difference between radians and degree is how you divide a circle in order to get sectors of one unit of angle

Semiclassical

Well, 1 rad is less than a third of pi radians

Abhas Kumar Sinha

why @Semiclassical If i can do cancel sin in eq. $\sin \theta = sin (90 - x)$ and not in $sin 45 deg = \sin \frac{\pi}{4}$?

Lozansky

2:44 PM

Can we, from this, conclude $|u(0,\theta)|<\infty$?

Semiclassical

What??

Abhas Kumar Sinha

For example in a equation (as an example) - $\sin x = \sin (90 - x)$, I can cancel sin both sides? or not? But. why I can't do cancellation here $\sin \frac{\pi}{4} radians = \45 deg$ here I can't cancel sin from both sides. Why?

Semiclassical

No, you cannot do so in general

Abhas Kumar Sinha

Why? My teacher did the first equation in the class

I asked that I didn't agree, but he said that we can do that

Semiclassical

There are cases where that will work, but it requires further conditions

Secret

2:48 PM

$\sin x = \sin (90 - x)$ is not the same as $x = 90 - x$ in general

Abhas Kumar Sinha

Even if we cancel the sin from both sides of any equation we don't get any wrong result

Semiclassical

For instance, it does work if you require x to be an acute angle

Abhas Kumar Sinha

In a right angle, x can't be 90 deg or greater

Secret

In fact, for the 'cancellation' to be valid, you will found that $x = 45$

Abhas Kumar Sinha

because of Sum angle property

Secret

2:49 PM

If you put $x$ as any other acute angle, then it falls apart

Abhas Kumar Sinha

@Secret that works for both equation

Secret

showing that that 'cancellation' is really a special case

Semiclassical

Sure. So for a right triangle you can’t run into problems with that

Secret

@Semiclassical Surely $\sin 30 \neq \sin (90-30) = \sin 60$?

Semiclassical

But you can take sine/cosine of angles larger than 90

Abhas Kumar Sinha

2:50 PM

Okay , @Secret now $sin x = \sin (3*x)$

that works

wooops that is - sign, not *

Semiclassical

@Secret it works in the sense that you can find a unique acute angle if you apply inverse sine to both sides

Abhas Kumar Sinha

@Secret Okay, if you take the value of sin as valid equation, then it works

cancellation I mean

@Semiclassical A right angled triangle can't have anything more than acute angles

Semiclassical

yes. But that’s not the full definition of sine/cosine

Abhas Kumar Sinha

Cancellation works every time, but why it doesn't work? Can anyone disprove cancellation

Secret

It really only works for one particular x. We don't call that cancellation. For it to be a cancellation, it has to work for all x

Let me demonstrate again:

You claim the following is valid:

$$\sin x = \sin (90 - x)$$

Abhas Kumar Sinha

2:55 PM

yea

Semiclassical

The full definition is done in terms of the unit circle and allows angles of arbitrary magnitude

Secret

and that you can "cancel out the sin"

Now suppose that is true, we will get:

$$\not\sin x = \not\sin (90 - x)$$
$$x = 90 - x$$

Abhas Kumar Sinha

yep

then?

Secret

But for that final equation, we can rearrange and get:

$$2x = 90$$

$$x = 45$$

Abhas Kumar Sinha

yea

Secret

2:56 PM

Meaning that it only works if $x = 45$

Abhas Kumar Sinha

yup

Semiclassical

You’re sorta missing the point. That’s the right solution in his case of interest @Secret

Secret

But for something to be cancellation, it has to always work, regardless of what value of $x$ is. This is clearly not true in the above case, hence it is not a cancellation

Abhas Kumar Sinha

Okay, gotcha, sin is not a number its something which gives a unique ratio, not number in algebraic form, so we can't cancel sin

right?

for example sin is not a variable, its a function to get ratios, If we cancel sin that that won't work

Semiclassical

That’s part of it, yeah. Sine is a function

Secret

2:59 PM

yes, sin is basically a machine that spits out a ratio when given a number, thus it is not a number as the outcome will change depending on what you feed into it

Abhas Kumar Sinha

Okay what are functions?

to give unique results on performing them on mathematics

Secret

A function is literally a black box that given some inputs (such as numbers) and spits out an output

Semiclassical

A function is a machine that sends an input to an output

Abhas Kumar Sinha

okay as in programming?

function () {}?

Tuki

similiar yes

Semiclassical

3:00 PM

So, for acute angles: Sine takes in an angle and outputs the corresponding ratio in a right triangle

Tuki

anyway how do computers compute decimals of sine ?

Semiclassical

Conveniently, for acute angles this mapping is 1-to-1

Tuki

taylor polynomial ?

Semiclassical

@Tuki nah

Abhas Kumar Sinha

@Secret okay, that's not cancellation, its something inside the input of functions to be performed before the calculation of values of function to get the value ? @Secret

Semiclassical

3:01 PM

There’s an algorithm which uses matrix multiplication

Tuki

$$ sin(x)=x-\frac{x^3}{3!}+\frac{x^2}{5!}-\frac{x^7}{7!}+\dots $$

Secret

@AbhasKumarSinha That is really an example of a special case where an incorrect method happened to give the correct answer

Semiclassical

I’m forgetting the name of it

Tuki

I'll try to look it up

Abhas Kumar Sinha

okay

Secret

3:03 PM

More precisely, $x = 45$ just happened to satisfy the equation $\sin x = \sin (90 -x) \implies x = 90 -x$

Semiclassical

en.m.wikipedia.org/wiki/CORDIC

There we go

Abhas Kumar Sinha

yep, here we work as an equation for the input of the functions, not the functions are cancelled?

not the output of the functions?

Lozansky

@Semiclassical Are you familiar with weighted $\delta$-distributions?

Semiclassical

There’s two issues here, one more subtle than the other

One is the terminology of ‘cancelling’ the operation

Abhas Kumar Sinha

Thanks @Semiclassical @Secret @Lozansky @tuki and everyone here!

Semiclassical

3:05 PM

As a way to refer to going from f(x)=f(y) -> x=y

But f(x) doesn’t refer to multiplication by f, so we avoid that language

Rather we say that we ‘invert’ sine. More precisely, we look for another function g(x) such that g(f(x))=x for all x.

Tuki

$\sin^{-1}(x)=\arcsin(x)$ also there is $\arcsin$ notation for inverse of $\sin(x)$

Semiclassical

g is then said to be the inverse function. In that case we have x=g(f(x)) = g(f(y)) = y

Abhas Kumar Sinha

sec

Semiclassical

So I’d say : “Applying inverse sine to both sides of the equation sin(x)=sin(pi/4-x), we get x=90-x.”

symmetric_cow

@abenthy I think this should be true... checking the injectivity of the map $G(\mathbb{Q}) \rightarrow G(\mathbb{R})$ amounts to checking that if you have maps $f,g: A \rightarrow \mathbb{Q}$ inducing the same map after composing the map $\mathbb{Q} \hookrightarrow \mathbb{R}$ (where $A$ is any $\mathbb{Q}$-algebra), then $f=g$, but I think this is just set theory..

Semiclassical

3:11 PM

There’s an additional subtlety beyond that, but that’s the terminology I’d usr

Balarka Sen

My copy of Milnor's Morse theory arrived today

Semiclassical

Back in a bit

Abhas Kumar Sinha

@Semiclassical how to calculate the values of a sine function, if the input is in degrees?

anyone knows how to do it?

How can I calculate the value of sinA, for a particular angle in degrees and get the ratio of perpendicular/hypotenuse for a particular triangle?

jublikon

3:27 PM

Hey guys, how's going?
I am stuck at the following task with complex numbers:

$\forall \in \mathbb{C}$ find $|1 + i z|^2<1$ and $Im(z-i)>0$

What I have done so far:

$z= x+iy, \quad x,y \in \mathbb{R} , i \in \mathbb{R}$

$
\begin{align}
|1+iz|^2<1 & \equiv |1+ i(x+iy)|^2 <1 \\
&\equiv |1+ix+y|^2 < 1
\end{align}
$

How could I proceed to find the solution set? Am I on the right track so far?

philmcole

@AbhasKumarSinha for a few certain values one can remember the values, e.g. 45°, 30°, 60°, 90° etc. For the rest you'll use a calculator. improveyourmathfluency.files.wordpress.com/2015/08/…

Abhas Kumar Sinha

@philmcole Is there anyway to do that without calculator ?

Yep I know those values

philmcole

Like $\sin(32.837°)$?

Abhas Kumar Sinha

But for other values, like 1, 2, 3..... 90?

yea

how to do that?

philmcole

No

Abhas Kumar Sinha

3:31 PM

?

what you mean?

philmcole

There is no closed formula for every value.

Abhas Kumar Sinha

okay, for sin(32.837)?

how you'll do that?

philmcole

type it in your calculator: wolframalpha.com/input/?i=sin(32.837%C2%B0)

Cookie Toast

Lord forgive me for my sin(s)

Abhas Kumar Sinha

So, there's no other way except calculator? So, how sin tables and calculators are made?

@philmcole

How to make my own Sin Table?

Cookie Toast

3:34 PM

@AbhasKumarSinha we can approximate trig functions as polynomial functions. Calculators can then use these to very very efficiently calculate trig functions to many decimal places

There are also some methods that allow you to calculate certain values of trig functions using different identities

Abhas Kumar Sinha

okay, what are approximate trig functions @CookieToast

Lozansky

@jublikon Well first of all $i(x+iy) = ix - y$

Akiva Weinberger

You'd use either trig identities (if you know $\sin(\theta)$ you can find $\sin(\theta/2)$) or use things like polynomial approximations

Abhas Kumar Sinha

Polynomial approximations @AkivaWeinberger

?

Cookie Toast

You probably won't be exposed to them until calculus, but, for example:
$\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \cdots + $

Abhas Kumar Sinha

3:36 PM

What are calcus? @CookieToast

Cookie Toast

Mathy stuff involving straight lines touching curves.

Akiva Weinberger

Calculus is a branch of math that deals with infinitesimals essentially

Cookie Toast

Someone else here would be able to give you a real definition. I don't know much

Abhas Kumar Sinha

So, Is there's no geometrical way or any other theoretical way?

What can I do with Calculus?

Cookie Toast

Using derivatives, you can derive the taylor polynomial above

Akiva Weinberger

3:37 PM

It's used all the time in physics

and for finding maxima and minima of functions

jublikon

@Lozansky oh... yes, stupid mistake.

Abhas Kumar Sinha

What can I do with Calculus?

Akiva Weinberger

11

Q: How were the sine, cosine and tangent tables originally calculated?

As I understand it... ahem... the (cosine, sine) vector was calculated for (30 degrees, PI/6), (45 degrees, PI/4) and (60 degrees, PI/3) angles etcetera, however, I would like know the original geometrical process for calculating the magnitudes for each vector in the trigonometric lookup table. ...

trigonometry

@AbhasKumarSinha Let's say you're moving at a constant velocity of 5 meters per second. After $t$ seconds, you've moved a distance of $5t$, right?

Abhas Kumar Sinha

okay

Akiva Weinberger

Now, what if your velocity is not constant, like at time $t$ your velocity is $t$ meters per second. So your velocity keeps on changing.

Abhas Kumar Sinha

3:42 PM

okay

Akiva Weinberger

It turns out that, after $t$ seconds, you'll have traveled $t^2/2$ meters. And you use calculus to figure that out.

Abhas Kumar Sinha

then average might work

Akiva Weinberger

So that's a very simple example of where it's used in physics.

Abhas Kumar Sinha

okay, gotcha, it seems quite difficult

Akiva Weinberger

Calculus is also used in optimization problems

Abhas Kumar Sinha

3:43 PM

But, do you think any possibility of the discovery of any simple method of calculating sin values without calculus?

Tuki

I would recommend to read about limits first

Abhas Kumar Sinha

limits?

Akiva Weinberger

What exactly would you consider to be a solution?

Tuki

Limit (mathematics)

In mathematics, a limit is the value that a function or sequence "approaches" as the input or index approaches some value. Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. In formulas, a limit is usually written as lim n → c f ...

Akiva Weinberger

Do you want a formula that just involves $x$, plus, minus, times, division, exponentiation, and roots?

Abhas Kumar Sinha

3:44 PM

yea

Tuki

Derivative is defined with limits

Abhas Kumar Sinha

@AkivaWeinberger

Akiva Weinberger

Then no such formula exists.

Abhas Kumar Sinha

Can any another formula be discovered?

to do that?

Akiva Weinberger

(Well, unless you allow yourself infinitely long formulas, but then you've stepped into the realm of calculus)

Abhas Kumar Sinha

3:45 PM

with those operations?

without using Calculus?

Tuki

$$ f'(a)= \lim_{h \rightarrow 0} \frac{f(a+h)-fa}{h} $$

Akiva Weinberger

Just with those operations, no. It turns out that sine is what's called a "transcendental function", and one of the consequences of being a transcendental function is that it can't be written as a finite combination of non-transcendental things

(such as plus, minus, times, division, exponentiation, and roots)

Abhas Kumar Sinha

No. I mean geometric Proof without functions and constructions

just using few triangles and circles?

Akiva Weinberger

I don't understand. A proof of what?

Abhas Kumar Sinha

and few other stuff?

Balarka Sen

3:47 PM

@AkivaWeinberger Is that not a consequence of Schanuel's conjecture.

I don't think it's a theorem

Semiclassical

It should be pointed out that historically the problem of finding sine for a range of different values was a challenge

Abhas Kumar Sinha

to prove a number that is the value of ratio of perpendicular/hypotenuse @AkivaWeinberger

for a given angle

Akiva Weinberger

@BalarkaSen Er… I doubt it?

@AbhasKumarSinha You can do it for some special angles

Abhas Kumar Sinha

then it might be possible to find other sin values for a given angle

Akiva Weinberger

Like, $\sin(60^\circ)=\frac{\sqrt3}{2}$

Abhas Kumar Sinha

3:49 PM

No, I mean all angles, I've something in mind, but I'm not sure that'd work

To do with triangles and circles

Lozansky

@AkivaWeinberger $\sqrt{3}/2$

Abhas Kumar Sinha

yep

Balarka Sen

@Akiva er yeah im thinking of something else. that's just because sin(1) is transcendental...

if you could write sin(z) as an algebraic function that would contradict

Akiva Weinberger

Oh, er, there is one technicality

The $\frac{e^{ix}-e^{-ix}}{2i}$ thing

Balarka Sen

yeah you need Lindemann-Weierstrass prolly

Akiva Weinberger

3:52 PM

I don't know if you've learned about complex numbers?

It turns out that there's a way to have a complex exponent

Like $2^i$

and if you allow that, than there is a simple way to write the sine function

Hrm. Actually, yeah, you don't even technically need the exponent to be a complex number.

$\dfrac{E^x-E^{-x}}{2i}$ where $E=\cos(1)+i\sin(1)$

I mean, you still have the question of how to raise a complex number to a possibly irrational exponent, but besides for that

if you allow complex numbers than it is possible

Balarka Sen

@Akiva learn symplectic geometry with me dawg

Akiva Weinberger

What even is that

Balarka Sen

So an inner product space is a vector space with a positive definitive, symmetric billinear form, yeah?

Akiva Weinberger

What's positive definite again?

Er, just $\langle x,x\rangle\ge0$?

Balarka Sen

Right-oh

Akiva Weinberger

3:58 PM

Okeydokes

Balarka Sen

@Alessandro trying hard to not get sniped

Alessandro Codenotti

@AkivaWeinberger And you have = only if $x=0$ (otherwise it's positive semidefinite)

@BalarkaSen I wasn't sure about which one should be transposed cause I always get it wrong so I just deleted the whole thing instead lol

Balarka Sen

lmao

Transcript for

$Mathematics$ Mathematics