New answers tagged

2

The key to excel in anything is sheer practice and hard work. Keep this thing in your mind whenever you suffer any setback at any stage of your life. Now for your advice on how to do well in mathematics, especially calculus. Well, since you say that you are a computer geek, I would recommend that the algorithm to follow is: First and foremost, take a ...


0

Set $t=0$, giving $$a\sin(0)+b\cos(0)=b=c\cos(f).$$ Then set $t=\frac\pi2$, and $$a\sin\left(\frac\pi2\right)+b\cos\left(\frac\pi2\right)=a=c\cos\left(\frac\pi2+f\right)=-c\sin(f).$$ This gives you the transfrom from $c,f$ to $a,b$. You can invert it by noting that $$a^2+b^2=c^2\cos^2(f)+c^2\sin^2(f)=c^2,$$ and $$\frac ...


3

$$ a\sin t+b\cos t=\sqrt{a^2+b^2}\left[\frac{b}{\sqrt{a^2+b^2}}\cos t +\frac{a}{\sqrt{a^2+b^2}}\sin t\right]$$ $$ =\sqrt{a^2+b^2}\left[\cos \alpha \cos t +\sin\alpha \sin t\right]$$ $$ =\sqrt{a^2+b^2} \cos(t-\alpha). $$ $ (c,f) $ have different symbols. Useful in combining two waves of same frequency but different amplitude.


0

We see that by the addition formula, $$c \cdot \cos(t + f) = c \cdot \left( \cos(t)\cos(f) - \sin(t)\sin(f)\right).$$ If you set $a = -\sin(f), c = 1,$ and $b = \cos(f),$ You can make a reasonable comparison between the expressions $a \sin(t) + b \cos(t)$ and $c \cdot \cos(t + f).$


0

Let's define $$ y(t)=\frac{a\omega\cos(\omega t)-b\sin(\omega t)}{a^2\omega^2+b^2} $$ $$\Omega={\omega\over a^2\omega^2+b^2}$$ Then, as shown in enzotib's answer, $y$ solves it on $[-\pi/\omega,0]$ and $-y$ on $[0,\pi/\omega]$. In $0$ they are $\pm a\Omega$ and their derivatives are $\mp b\Omega$. But then we also have the $e$ thingy, which will help us ...


1

Given that the right-hand side is not differentiable in the points $k\pi/\omega$, you could only find the general solution, separately, on each interval not containing one of these points. For example, in $(0,\pi/\omega)$ the solution is $$ x(t)=c_1 e^{-\frac{b}{a}t}-\frac{a\omega\cos(\omega t)-b\sin(\omega t)}{a^2\omega^2+b^2} $$ while in $(-\pi/\omega,0)$ ...


5

Have you ever felt self doubt in your career as a mathematician? How did you overcome those worries? Also, what are some good techniques or resources to advance one's skills as a undergraduate level mathematician? You bet I did. I graduated with a B.S. Mathematics, Statistics emphasis two years ago, and I am now pursuing a M.S. Statistics at a top-20 ...


5

Im not studying and Im not going to study mathematics officially (Im a bit "old"), Im an amateur. When started to follow my hobby more seriously sometimes I feel that Im not understanding anything just memorizing things. But the brain is a giant mystery: one or two years after I started the "hobby", with long periods of time not seeing something about ...


8

Even some of the most extraordinarily talented mathematicians, scientists, and scholars experience doubt--personal doubt that they've chosen the right career, worked on a solvable or important problem, that they can ever solve the problem (before a competitor) and so forth. This happens with every challenging discipline. Judge your progress both on a ...


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disclaimer: I'm also a sophomore in college studying mathematics. I've taken algebra, analysis I, measure theory, algebraic topology. Here are some things that helped me a great deal: (1) Spend a lot of time here, you get a sense for how proofs go, in some sense. It's very worthwhile to see how different people approach a singular problem. Answering ...


0

Pythagoras' theorem! Proof using a rotated square within a square. This gets straight to the essence of mathematics!


0

The general method to solve such kind of problems is to take a point $P=(x,y)$ and write the given conditions. For 1) we have the point $F=(2,3)$ and the line $d$ of equation $x=1$ and we want that the distance $\overline {Pd}$ is equal to the distance $\overline{PF}$. You can easily that $\overline {Pd}=|x-1|$ and, using the formula for the distance : ...


2

Everybody's talking about radians vs degrees. What about radians vs turns, a turn being $2\pi$ radians? I'm asking because it seems logical to think of a measure of angle as the fraction of a full circle, also I end up putting so many $2\pi$'s everywhere (mostly in sound processing) that it makes me wonder why not go the other way and use turns. The worst it ...


2

Radians tell you the arc length. If you have 60° you then have a bit of work to figure out the arc length: Al = n°/360 * 2πr but if you have π/3 radians, you know that the arc length is π/3 radii or π/3 * r


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I don't know if this answer is good or not. Reason 1 Lets say you want to measure the distance of a very distant star from earth like in the below image. Consider that small circle earth and the big one, some distant star. Say you observe the star from two different points on earth. Now you can find the angle $\alpha$ and $\beta$ approximately and you ...


3

it's because calculus would be really annoying using degrees. $$ (\sin x)' = \cos x. $$ in radians but not in degrees. Also, the related fact $$ \cos x = \frac{e^{ix}+e^{-ix}}{2} $$ holds in radians.


1

As others noted, radians are a natural choice in mathematics. However, the same reason Babylonians chose 360 as the number of degrees in a circle (nice subdivisions of the whole circle) makes 360 a better choice for a full circle in numeric applications intended for graphics than $2\pi$: every different floating point format has a different numerical ...


5

radians are the natural unit of measure for angles. it's no anthropocentric convention. aliens on the planet Zog that do calculus and solve physics problems will also be understanding the naturalness of describing angles in radians. as mentioned previous, the angle, expressed in radians, is the amount of circular arc swept by the angle divided by the ...


4

Degrees are a mistake of history (not speaking of minutes and seconds). Division in four quadrants of ninety degrees is quite arbitrary and inconvenient, but for one thing: it allows an easy representation of the remarkable angles, $30°$ and $45°$. In this respect, it is a little better than the $4\times100$ subdivisions in grades. As explained by many ...


1

(Note: See edits below original post.) Here is another reason why: Radians are unitless (a.k.a. dimensionless). This means that "$2\pi$ radians" actually equals $2\pi \ $ (the number) without any need to signify a unit of measurement. With $360^\circ$, it is absolutely necessary that we include the "$^\circ$" symbol to signify that it is in the unit ...


0

The answer is simple, it's a distance measure. When you move in a straight line you use inches or metres, in circles it is radians. If you are at Disneyland and ask how far it was to Anaheim Stadium [go, Angels!] and I tell you that from my house it's about 45º, you are probably not going to be happy. You want the distance traveled, at 1 mile out from my ...


1

Radians are in some sense the "natural" units in which to measure angles. For a circle of radius r, an angle of A radians will subtend an arc on that circle of length rA. The use of degrees or grads is just a change of units for measuring angles but if one uses units other than radians, one must always carry around conversion factors like (180/\pi) all ...


2

The simple reason is that radians incorporate pi as part of the ratio which tends to be more convenient for arbitrary calculations and more complex mathematical functions. For strictly practical manufacturing type applications degrees are often preferred because they are easier to visualise and subdivide. In engineering radians tend to be used for ...


102

The reasons are mostly the same as the fact that we usually use base $e$ exponentiation and logarithm. Radians are simply the natural units for measuring angles. The length of a circle segment is $x\cdot r$, where $x$ is the measure and $r$ is the radius, instead of $x\cdot r\cdot \pi/180$. The power series for sine is simply ...


6

Radians naturally arise when you look at some circles (note that they are a dimensionless unit). On the contrary, full circle being $360^\circ$ is due to some dude dividing the circle to as many pieces as there are days in the year (for some historical reason this resulted in $360$). Why people think in radians then? My personal guess is that the reason is ...


63

As I teach my trigonometry students: "Degrees are useless." You want to know the length of a circular arc? It's $r \theta$ where $r$ is the radius of the circle and $\theta$ is the angle it subtends in radians. If you use degrees, you get ridiculous answers. You want to know the area of a sector? It's $\frac{1}{2} r^2 \theta$, with $r$ and $\theta$ as ...


0

$$ \sin\left(x-\frac{2}{3}\pi\right)=-\sin\left(\frac{2}{3}\pi-x\right)= $$ $$ =-\cos\left(\frac{2}{3}\pi-x-\frac{\pi}{2}\right)=-\cos\left(\frac{4}{6}\pi-x-\frac{3\pi}{6}\right)=-\cos\left(\frac{\pi}{6}-x\right) $$ where I only used the facts that $\sin\left(-x\right)=-\sin\left(x\right)$ and $\sin\left(x\right)=\cos\left(x-\frac{\pi}{2}\right)$


3

Use $$\sin\left(x\right)=\cos\left(\frac{\pi}{2}-x\right)$$ $$\cos (\pi+x)=-\cos x$$ The result follows.


1

Such a parallelogram does not exist, as the height is greater than the adjacent side length. Hence you can't draw


-6

I have a beautifully simple argument which will help you to show many doubters the true situation. If you project the real number line on to a circle it is clear that the real numbers are dense everywhere, while the natural numbers are dense only at infinity. This creates a strong, visualisable argument that there are more real numbers than integers. The ...


1

This might help. The terms "addition/multiplication/division" really don't have anything to do with the definition of groups. They are all just aliases for some familiar binary operations. Mostly they influence our notation for the operation of the group (so that it is juxtaposition, a dot, or +, or something else.) A groups operation and its elements' ...


1

I guess your question should probably be : why do we not study monoids first (monoids are like groups, but without the hypothesis that elements have an inverse) ? And a reason is that, in addition to the fact that groups are usually considered more "important" than mere monoids, the study of monoids is actually more complicated than the study of groups. The ...


0

I am unfamiliar with anyone but a single nutjob that would go about division rather than multiplication first. Notationally however one may use division, in a group with multiplicative notion we might write $a/b$ to mean $a\cdot b^{-1}$ as nothing but a shorthand. Additive notion for groups is usually reserved for abelian groups.


0

Historically speaking, things really took off in analysis in the 17th century when the concept of change received a mathematical form with the development of infinitesimal calculus by the likes of Fermat, Barrow, Leibniz, and Newton. So possibly the concept of change is even more important historically than that of function which did not take center stage ...


0

I enjoyed this one very much: How to compute $\sum 1/n^2$ by solving triangles


4

Let the area of the triangle be $\Delta$, and the inradius be $r$ Then dividing the triangle into three by joining each vertex to the incentre we get $r(a+b+c)=2\Delta$. The area of the circle is $\pi r^2$. So all we need is the formula for $\Delta$ which is $\Delta=\sqrt{s(s-a)(s-b)(s-c)}$, where $s=(a+b+c)/2$. With $a=15,b=14,c=13$, we have $s=21$, so ...


1

You don't want a "proof without words", because there is nothing to prove here, but an access to logarithms "for a younger audience". Begin with the table $$\eqalign{100&\quad{\rm has\ 2\ factors\ of\ 10}\cr 1000&\quad{\rm has\ 3\ factors\ of\ 10}\cr 10\,000&\quad{\rm has\ 4\ factors\ of\ 10}\cr 100\,000&\quad{\rm has\ 5\ factors\ of\ ...


-1

OK, I followed a trail and came out to this page: apparently a proof without words is a picture that visualizes a theorem thesis. My first reaction is that visualizing is not the only way to make a child understand. I'd say that using words is just as noble and effective as that. Words only? Well, unfortunately or not, in maths symbols are just beyond the ...


0

There is a youtuber who also works at Khan Academy as far as I am aware. He created his own animation program specifically designed for mathematics animation and I believe anyone is free to use it. I think it looks very good. He goes by the username 3Blue1Brown on youtube.


0

One option is also GNU Dr. Geo, a libre interactive geometry and programming software I wrote, you can use it on Mac OS X Regarding plotting capabilities, it is best used with its programming interface to produce very interesting and appealing interactive diagram. For example, related to single variable function, root finding algorithms can be compared. ...


0

This may be more of a niche motivation and is meant to complement rather than replace the above, but for those economics students who like economic theory, linear algebra is important because the a great deal of math that is interesting to economic theory eventually becomes linear algebra. Algebraic topology is clearly relevant (if a topic economics have ...


0

I really enjoyed Bergman's supplement to the exercises in Rudin's Principles. He also has a short set of notes to instructors and answers to questions from students. These helped me a lot in going through Rudin. Also, I don't think they're at a disadvantage if you do not provide them with solutions. There are plenty to be found on the web if they want to ...


1

There exists a certain variation-or rather "enrichment"-of the Braille Alphabet, named Nemeth Braille, after its' creator, Abraham Nemeth, which is also using the standard six-dot Braille cells to create mathematical symbols. I am not sure on whether it is exhaustive-that is, if all mathmematical expressions can be written by making use of its symbols-but I ...


2

It is DeMoivre's rule and you stated it wrong. $$ \left(\cos \theta + i \sin \theta \right)^n = \cos n \theta + i \sin n \theta.$$ If $n\theta$ is a multiple of $\pi$ the end result is $\pm 1$, not $0$.


2

This is a very good question. Most responses to the video instruct the viewer to use zeta function regularization or Ramanujan summation, without actually engaging the proof and showing why it's wrong. Or their demonstrations why it's wrong ignore the entire body of summability methods for divergent series. Although they are not careful in the video to ...


10

Regarding math at the graduate level and beyond, these programming debugging techniques can be quite effective: Rubber duck debugging: Find someone/something to whom to explain the bug and your solution. This is also effective for helping undergraduates find bugs in their solutions to word problems. Code review, i.e., presenting your argument to an at ...


5

One approach which might interest you is Automated proof checking, where you write the proof in a form that a computer can check the validity of. It is not easy to write proofs in this form, as humans tend to skip lots of "elementary" steps when writing proofs for other humans to read. A computer-checkable proof will be much longer and more tedious. ...


6

Another method is to apply your method to other inputs that leads to larger errors. The idea is that the larger the error, the more obvious it becomes where the error is coming from. So, instead of 3 T's where the error is just a factor 2, if you consider, say, 100 T's then the error is astronomically larger, making the source of the error much more ...


17

I'm a student currently learning math at the undergraduate level. I mostly learn by self-study, so "debugging" my thinking process during the learning process is crucial. (I will interpret "debug" a little less literally, so more as checking your own work using alternate solutions / shortcuts.) This list is not exhaustive, but off of the top of my head here ...


7

There are two strategies I could use to debug this problem. One is to write the steps out one at a time, with more detail. Let's use this as an example: Claim: There are 24 ways to order 3 Ts and 1 H if the Ts are distinguishable Claim: There are 3 Ts, so if the Ts are not distinguishable, each order is counted 3 times because each one is counted 3 ...



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