I'm currently learning about hyperbolic trig functions, and i don't really get the point. At first, I found it really weird that the input is the area divided by 2, and just wondered what was the point of it anyways?  So, are there any real world applications for this kind of stuff?
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I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to why (or even if) they are useful.  Is there any good examples of their uses outside academia?
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I have two points. I need to draw an arc ($&lt;180$°) between them, and I know how long it should be, but nothing else about it.   Knowing either the radius length or the coordinates of the center point of the circle should be enough to draw it directly. I could also calculate the radius if I knew the subtended angle θ. But I don't know how to calculate either of these.   I know that somebody answered a  a while ago. But for some reason, the formula they give for the answer includes θ. Maybe there is an easy way to calculate it and my knowledge is just too rusty, but I can't solve my problem with this information.   I tried deriving an answer by myself, but arrived at an equation containing both an angle's cosine and the angle squared, and I don't know how to solve such an equation. Besides, there is probably an easier way than what I used.   It is OK if the answer is derived by using trigonometry, analytical geometry or anything else, as long as I can use a pocket calculator to get my radius/coordinates. But if you don't want to lose me along the way, it would be nice if you could use calculations at undergraduate level.   Here is a drawing of what I mean:
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I read this in a math textbook, stated as obvious fact. I cannot wrap my head around why this works. I understood the division lemma, and also got the algorithm to work: but the proof of why the algorithm works, just said "because of the equation" that I put in the question.  Could you please elaborate on why it works?
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If you take the derivative of the volume of a sphere, $$\frac{4}{3}\pi r^3$$you get its surface area, $$4\pi r^2.$$ If you differentiate again, you get $$8 \pi r.$$ Does this have any physical (or other kind of) significance, besides being 4 times the length of a great circle on the sphere?
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What are the exact values of $\cos(2\pi/7)$ and $\sin(2\pi/7)$ and how do I work it out?  I know that $\cos(2\pi/7)$ and $\sin(2\pi/7)$ are the real and imaginary parts of $e^{2\pi i/7}$ but I am not sure if that helps me...
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Why is $\pi r^2$ the surface of a circle?   I have learned this formula ages ago and I'm just using it like most people do, but I don't think I truly understand how circles work until I understand why this formula works.   So I want to understand why it works and not just how.  Please don't use complicated symbols.
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So I am looking at the integral $$ \int_0^{2\pi} \frac{1}{a + \cos\theta} \mathrm{d} \theta$$ for $a &gt; 1$. Evaluating the integral with complex analysis gives us $\frac{2 \pi}{\sqrt{a^2 - 1}} $ but I have failed to reproduce the result using standard substitution.  Here’s what I did: I found the antiderivative first, which is $$\frac{2}{\sqrt{a^2 - 1}}  \arctan \left(\sqrt{\frac{a - 1}{a + 1}} \tan\left(\frac{\theta}{2}\right) \right)$$ but substituting in the bounds would give us 0.   I used the standard substitution $t = \tan\left(\frac{\theta}{2}\right) $ to obtain the following result.  EDIT: Plotting the graph shows us that the area is definitely positive, so there’s no way the answer is $0$.  EDIT2: Is it because $2\pi$ corresponds to the second “cycle” so we should $\arctan 0$ to be $\pi$ for our upper bound instead?
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I'm struggling to understand why the range of tangent function is equals to all real numbers. Because tangent = opposite/adjacent right? But the sum of the opposite^2 + adjacent ^2 must be equal to the hypotenuse^2. So doesn't that put a limit on the value of denominator (and the numerator) of the ratio/tangent, therefore putting a limit on the range of the function?   Also, I don't get how the tangent function is able to produce every single real number. Because doesn't the fact that opposite^2 + adjacent ^2 must be equal to the hypotenuse^2 prevent the tangent function from producing certain numbers? Because for any given opposite value, the adjacent value can't just be any integer. It's square must add up with the square of the opposite to equal hypotenuse squared. Doesn't that prevent the tangent function from producing certain numbers, because there are only so many combinations of opposite values and adjacent values that can occur?   Can you please explain to me why all possible real numbers can by formed by the tangent, even with these limits? Can you please explain why this is not the case as simply as possible, without using calculus (because I haven't learnt it)? I realize when you plot it, it will go up to infinity. But I want to understand why this is the case, without using calculus
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One thing, I'm not a mathematician so please be patient. I am still in Algebra II Trig. Leading with that, why does  $$ x_0 = \sin 1, \space  x_1 = x_0 + \sin x_0, \space x_2 = x_1 + \sin x_1 ... $$ and after a while, $$ x = \pi $$  I know this to be true because I have evaluated this on my TI-84 and more deeply evaluated it with this program I made:   To $100000$ Iterations (which I believe is enough)   So what's the math behind it? Why does adding $\sin 1$ in this manner produce $\pi$? Why doesn't doing the same thing using $100$ produce $\pi$? For example,  $$\sin(1) = 0.841470...,\space \sin(1) + \sin(0.8414...), \sin(1) + \sin(0.841470...) + \sin(1.587095126...),\space ... $$ $$ = \pi$$  My question is different from the mentioned because my function is not taking the sin of a sin consecutively, but is this following function: $f(x) = x + \sin x$ and is not $f(x) = \sin x$ and therefore can exhibit different properties I need explained.
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I've been wondering, what is the general form for the integral of $\sin(\cos x)$? I tried integrating it myself and couldn't, tried with Wolfram Alpha and it couldn't either. If you think about it conceptually, it should have a general form as it's graph if just like sin and cos, which both have general forms. If it doesn't have a general form, I'd like to know why. Thanks!
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Before I am told, I want to clarify that I searched first, and I don't believe this to be a repost. I understand the formula in terms of how to apply it, and I've seen graphical representations and everything. I get that we are finding where the tangent line has a root, then choosing a new $f(x)$ at that point and finding the root of its tangent line, effectively closing the distance between x and the root r.   What I do not understand, is what $\frac{f(x)}{f'(x)}$ is actually doing. I know it can be used to find the root x, as it is derived from $y=mx+b$, but how is dividing $f(x)$ by its derivative getting me the root? Why does this work? My intuition is telling me (before I actually tried it) that I was getting some y value, then seeing how many times the slope goes into it; but this would give me the $x$ coordinate, wouldn't it? I can use it, but it's not clicking as to why, and I'd like to fix that so I can actually understand what is going on.
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I don't understand this. These identities are given in the online notes for MIT's 18.01 calculus class. It's related to taking the sum of two trig functions and transforming them into a single trig function. I can use the formulas to do this, but I am having trouble finding anything on the internet to use a proof for my understanding.  Thanks for any responses!  EDITED (added after the first answer):  Sorry, the identity goes like this:  $A\sin{k(x-c)}=a\sin{kx}+b\cos{kx}$ ,  with the relationships  $a=A\cos{kc}$ , $b=-A\sin{kc}$ , $A=\sqrt{a^2+b^2}$ , and $\tan{kc}=-b/a$
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Do we rely on certain intuition or is there an unofficial general crude checklist I should follow?  I had a friend telling me that if the sum of the powers on the numerator is smaller then the denominator, there is a higher chance that it may not exists. And if the sum of powers on the numerator is higher then the denominator, most likely, it exists. Also if there is a sin or cos or e-constant, it most likely exists.  How can I know what to do at the first glance given so little time exists for me in exam to ponder? If I spend all my time on figuring out a two-path test when the limit exists, that would be a huge disaster. Is this one of those cases where practice makes perfect? Example: $$\lim_{(x,y)\to(0,0)}\frac{(\sin^2x)(e^y-1)}{x^2+3y^2}$$ Please give me a hint and where do you get the hint. Example: $$\lim_{(x,y)\to(0,0)}\exp\left(-\frac{x^2+y^2}{4x^4+y^6}\right)$$ I need a hint for this too.  Common methods I have learnt for reference: Two-Path test, Polar Coordinates, Spherical Coordinates, Mean Value Theorem using inequalities.
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Why does the addition of a \left and \right introduces an extra space around the formula? Specifically, why do $\cos(\theta)$ and $\cos\left(\theta\right)$ render differently?  Is there a way to use resizable delimiters without introducing the spurious spaces?  Why did Knuth design it like this?  Edit: This occurs too, if \operatorname{MYOP} or some other operator, like \lim, \min etc. is used. (This edit is basically to make it easier to find this answer.)
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Don't let the simplicity of this diagram fool you. I have been wondering about this for quite some time, but I can't think of an easy/smart way of finding it.  Any ideas?    For reference, the Area is:  $$\bbox[10pt, border:2pt solid grey]{90−18.75\pi−25\cdot \arctan\left(\frac 12\right)}$$
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I started learning about the  in my calculus class, and although I understand the material well enough, I'm not really sure what actual applications there are for the series.  Question: What are the practical applications of the Taylor Series? Whether it's in a mathematical context, or in real world examples.
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Okay I like have no idea how to do this. I tried integration by parts then everything came back uglier. I then put this in wolfram it says "No found in standard mathematical functions"   Is there a easy way to do this?   Maybe like FTOC II?