Trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles, and other topics relating to measuring triangles.

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33
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2answers
2k views

Integral $\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$$ It also can be represented as $$I=\int_0^{\pi/4}\frac{\phi^2}{\cos \phi\,\sqrt{\cos ...
46
votes
5answers
3k views

Why does $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$?

Playing around on wolframalpha shows $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$. I know $\tan^{-1}(1)=\pi/4$, but how could you compute that $\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi$ to get this ...
14
votes
3answers
482 views

Prove that $\int_0^\pi\frac{\cos x \cos 4x}{(2-\cos x)^2}dx=\frac{\pi}{9} (2160 - 1247\sqrt{3})$

Prove that $$\int_0^\pi\frac{\cos x \cos 4x}{(2-\cos x)^2}dx=\frac{\pi}{9} (2160 - 1247\sqrt{3})$$ I tried to use Weierstrass substitution but the term $\cos 4x$ made horrible algebraic-forms since ...
19
votes
4answers
2k views

Is there a limit of cos (n!)?

I encountered a problem today to prove that $(X_n)$ with $X_n = \cos(n!)$ does not have a limit (when $n$ approaches infinity). I have no idea how to do it formally. Could someone help? The simpler ...
16
votes
2answers
1k views

De Moivre's Theorem. Motivation and origins.

I've purchased "A Source Book in Mathematics" some time ago and I'm still baffled by De Moivre's paper on his formula. We all know the famous $$\{\cos(x) + i \sin(x)\}^n = \cos(nx)+i \sin(nx)$$ but ...
14
votes
2answers
820 views

Proving $2 ( \cos \frac{4\pi}{19} + \cos \frac{6\pi}{19}+\cos \frac{10\pi}{19} )$ is a root of$ \sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$

How can one show that the number $2 \left( \cos \frac{4\pi}{19} + \cos \frac{6\pi}{19}+\cos \frac{10\pi}{19} \right)$ is a root of the equation $\sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$?
8
votes
3answers
994 views

When the trig functions moved from the right triangle to the unit circle?

I have to write a paper about the unit circle and I'm trying to uncover some of its origins. Also, when the trig functions were expanded to angles greater than 90° and what was the rationale behind ...
13
votes
4answers
3k views

Do “imaginary” and “complex” angles exist?

During some experimentation with sines and cosines, its inverses, and complex numbers, I came across these results that I found quite interesting: $ \sin ^ {-1} ( 2 ) \approx 1.57 - 1.32 i $ $ \sin ...
10
votes
3answers
758 views

Intuition of Addition Formula for Sine and Cosine

The proof of two angles for sine function is derived using $$\sin(A+B)=\sin A\cos B+\sin B\cos A$$ and $$\cos(A+B)=\cos A\cos B-\sin A\sin B$$ for cosine function. I know how to derive both of the ...
5
votes
1answer
310 views

Ramanujan-type trigonometric identities with cube roots, generalizing $\sqrt[3]{\cos(2\pi/7)}+\sqrt[3]{\cos(4\pi/7)}+\sqrt[3]{\cos(8\pi/7)}$

Ramanujan found the following trigonometric identity \begin{equation} \sqrt[3]{\cos\bigl(\tfrac{2\pi}7\bigr)}+ \sqrt[3]{\cos\bigl(\tfrac{4\pi}7\bigr)}+ \sqrt[3]{\cos\bigl(\tfrac{8\pi}7\bigr)}= ...
19
votes
8answers
3k views

Rapid approximation of $\tanh(x)$

This is kind of a signal processing/programming/mathematics crossover question. At the moment it seems more math-related to me, but if the moderators feel it belongs elsewhere please feel free to ...
12
votes
4answers
659 views

Why do we use the Euclidean metric on $\mathbb{R}^2$?

On the train home, I thought I would try to prove $\pi$ is irrational. I needed a definition, so I used: $\pi$ is the area of the unit circle. But what is a circle? A circle is the set of tuples ...
17
votes
4answers
2k views

How does e, or the exponential function, relate to rotation?

$e^{i \pi} = -1$. I get why this works from a sum-of-series perspective and from an integration perspective, as in I can evaluate the integrals and find this result. However, I don't understand it ...
8
votes
3answers
255 views

A trigonometric equation

Some years ago, I came across the following question: Find the value of $\tan5°\tan55°\tan65°\tan75°$. The value is 1 and I generalized the question as follows. $\tan a°\tan b°\tan c°\tan d° ...
7
votes
1answer
2k views

How were Hyperbolic functions derived/discovered?

Trig functions are simple ratios, but what does Cosh, Sinh and Tanh compute? How are they related to euler's number anyway?
4
votes
5answers
5k views

Good book recommendations on trigonometry

I need to find a good book on trigonometry, I was using trigonometry demystified but I got sad when I read this line: Now that you know how the circular functions are defined, you might wonder how ...
2
votes
1answer
151 views

A question by Ramanujan about a relational expression of a triangle

I found the following question in a book without any proof: Question : Suppose that each length of three edges of a triangle $ABC$ are $BC=a, CA=b, AB=c$ respectively. If $$\frac1a=\frac1b+\frac1c, ...
8
votes
4answers
267 views

Explaining $\cos^\infty$

I noticed something odd while messing around on my calculator. $$\lim_{x\to \infty} \cos^x(c)=0.7390851332$$ Where $c$ is a real constant. My calculator is in radians and I got this number by simply ...
7
votes
2answers
115 views

Method of proof of $\sum\limits_{n=1}^{\infty}\tfrac{\coth n\pi}{n^7}=\tfrac{19}{56700}\pi^7$

The following formula was stated by Ramanujan: $$\sum\limits_{n=1}^{\infty}\frac{\coth n\pi}{n^7}=\frac{19\pi^7}{56700}$$ Does anybody know the method of proof of this formula? I know that typically ...
7
votes
1answer
2k views

A series expansion of $\cot (\pi z)$

How to show the following identity holds? $$ \displaystyle\sum_{n=1}^\infty\dfrac{2z}{z^2-n^2}=\pi\cot \pi z-\dfrac{1}{z}\qquad |z|<1 $$
6
votes
1answer
943 views

How to calculate a heading on the earths surface?

Given an initial position and a subsequent position, each given by latitude and longitude in the WGS-84 system. How do you determine the heading in degrees clockwise from true north of movement?
5
votes
3answers
172 views

minimum value of $\cos(A-B)+\cos(B-C) +\cos(C-A)$ is $-3/2$

How to prove that the minimum value of $\cos(A-B)+\cos(B-C) +\cos(C-A)$ is $-3/2$
5
votes
3answers
6k views

Determine third point of triangle when two points and all sides are known?

Determine third point of triangle (on a 2D plane) when two points and all sides are known? A = (0,0) B = (5,0) C = (?, ?) AB = 5 BC = 4 AC = 3 Can someone ...
5
votes
1answer
2k views

Solving a trignometric equation of form $a\sin x + b\cos x = c$

Suppose that there is a trignometric equation of the form $a\sin x + b\cos x = c$, where $a,b,c$ are real and $0 < x < 2\pi$. An example equation would go the following: $\sqrt{3}\sin x + \cos x ...
22
votes
5answers
4k views

Why is $\sin(d\Phi) = d\Phi$ where $d\Phi$ is very small?

I haven't touched Physics and Math (especially continuous Math) for a long time, so please bear with me. In essence, I'm going over a few Physics lectures, one which tries to calculate the Force ...
11
votes
5answers
2k views

Why aren't the graphs of $\sin(\arcsin x)$ and $\arcsin(\sin x)$ the same?

(source for above graph) (source for above graph) Both functions simplify to x, but why aren't the graphs the same?
10
votes
2answers
3k views

What is the optimum angle of projection when throwing a stone off a cliff?

You are standing on a cliff at a height $h$ above the sea. You are capable of throwing a stone with velocity $v$ at any angle $a$ between horizontal and vertical. What is the value of $a$ when the ...
5
votes
1answer
673 views

Geometric construction of hyperbolic trigonometric functions

If we have a circle we can geometrically construct the trigonometric functions as shown. The functions all derive from sin and cos. If we say that the circle is a conic section and imagine it on the ...
5
votes
4answers
645 views

Prove that $\cot^2{(\pi/7)} + \cot^2{(2\pi/7)} + \cot^2{(3\pi/7)} = 5$

Prove that $\cot^2{(\pi/7)} + \cot^2{(2\pi/7)} + \cot^2{(3\pi/7)} = 5$ . I am sure this is derived from using roots of unity and Euler's complex number function, but I am very uncomfortable in these ...
4
votes
4answers
342 views

A trigonometric identity: $(\sin x)^{-2}+(\cos x)^{-2}=(\tan x+\cot x)^2$

I've been trying to prove it for a while, but can't seem to get anywhere. $$\frac{1}{\sin^2\theta} + \frac{1}{\cos^2\theta} = (\tan \theta + \cot \theta)^2$$ Could someone please provide a valid ...
3
votes
2answers
216 views

Find this limit without using L'Hospital's rule

I have to find this limit without using l'Hôspital's rule: $$\lim_{x\to 0} \frac{\alpha \sin \beta x - \beta \sin \alpha x}{x^2 \sin \alpha x}$$ Using L'Hôspital's rule gives: ...
3
votes
2answers
577 views

Proving: $\cos A \cdot \cos 2A \cdot \cos 2^{2}A \cdot \cos 2^{3}A … \cos 2^{n-1}A = \frac { \sin 2^n A}{ 2^n \sin A } $

$$\cos A \cdot \cos 2A \cdot \cos 2^{2}A \cdot \cos 2^{3}A ... \cos 2^{n-1}A = \frac { \sin 2^n A}{ 2^n \sin A } $$ I am very much inquisitive to see how this trigonometrical identity can be ...
2
votes
3answers
340 views

If $\sin(a)\sin(b)\sin(c)+\cos(a)\cos(b)=1$ then find the value of $\sin(c)$

If $$\sin(a)\sin(b)\sin(c)+\cos(a)\cos(b)=1,$$;where abc are the angles of the triangle.! then find the value of $\sin(c)$. By trial and error put this triangle as right angled isosceles and got the ...
2
votes
4answers
660 views

How to derive inverse hyperbolic trigonometric functions

$e^{i\theta}=\cos\theta + i\sin \theta$ $e^{i\sin^{-1}x}=\cos(\sin^{-1}x)+i\sin(\sin^{-1}x)$ $i\sin^{-1}x=\ln|\sqrt{1-x^2} + ix|$ $\sin^{-1}x=-i\ln|\sqrt{1-x^2} + ix|$ Now from here I'm kind of ...
2
votes
0answers
255 views

Product of Sine: $\prod_{i=1}^n\sin x_i=k$

From the article Products of Sines, we have $\sin 15^\circ\sin75^\circ=\sin 18^\circ\sin54^\circ=\frac{1}{4}$. We can rewrite this as $\sin \frac{\pi}{12}\sin\frac{5\pi}{12}=\sin ...
0
votes
1answer
445 views

How to prove $ \sin x=…(1+\frac{x}{3\pi})(1+\frac{x}{2\pi})(1+\frac{x}{\pi})x(1-\frac{x}{3\pi})(1-\frac{x}{2\pi})(1-\frac{x}{\pi})…$? [duplicate]

Possible Duplicate: infinite product of sine function Here is an other one which is more or less what Euler did in one of his proofs. The function sinx where x∈R is zero exactly at ...
9
votes
6answers
468 views

Derive $\frac{d}{dx} \left[\sin^{-1} x\right] = \frac{1}{\sqrt{1-x^2}}$

Derive $\frac{d}{dx} \left[\sin^{-1} x\right] = \frac{1}{\sqrt{1-x^2}}$ (Hint: set $x = \sin y$ and use implicit differentiation) So, I tried to use the hint and I got: $x = \sin y$ ...
6
votes
1answer
445 views

Rigorous proof of an infinite product.

I'll give a proof of the following expansion: $$\frac{\sin x}{x} = \prod_{i=1}^{\infty} \cos \frac{x}{2^i}$$ $${\sin x} = 2 \cos \frac{x}{2}\sin \frac{x}{2}$$ $${\sin x} = 2^2 \cos \frac{x}{2}\cos ...
5
votes
4answers
148 views

If $\frac{\sin^4 x}{a}+\frac{\cos^4 x}{b}=\frac{1}{a+b}$, then show that $\frac{\sin^6 x}{a^2}+\frac{\cos^6 x}{b^2}=\frac{1}{(a+b)^2}$

If $\frac{\sin^4 x}{a}+\frac{\cos^4 x}{b}=\frac{1}{a+b}$, then show that $\frac{\sin^6 x}{a^2}+\frac{\cos^6 x}{b^2}=\frac{1}{(a+b)^2}$ My work: $(\frac{\sin^4 x}{a}+\frac{\cos^4 ...
5
votes
3answers
4k views

How to find roots of $X^5 - 1$?

How to find roots of $X^5 - 1$? (Or any polynomial of that form where $X$ has an odd power.)
4
votes
4answers
246 views

Solving a trigonometric equation

How can I solve this trigonometric equation? $$\sin(x)=\tan(\frac{\pi}{15})\tan(\frac{4\pi}{15})\tan(\frac{3\pi}{10})\tan(\frac{6\pi}{15}).$$
3
votes
2answers
482 views

How to prove $\cos 36^{\circ} = (1+ \sqrt 5)/4$?

Given $4 \cos^2 x -2\cos x -1 = 0$. Use this to show that $\cos 36^{\circ} = (1+ \sqrt 5)/4$, $\cos 72^{\circ} = (-1+\sqrt 5)/4$ Your help is greatly appreciated! Thanks
3
votes
4answers
266 views

Given that $\;\sin^3x\sin3x = \sum^n_{m=0}C_m\cos mx\,,\; C_n \neq 0\;$ is an identity . Find the value of n.

Problem : Given that $\sin^3 x \sin 3x = \sum^n_{m=0}C_m \cos mx, C_n \neq 0 $ is an identity. Find the value of n. I tried : $\sin3x = 3\sin x - 4\sin^3 x$ but unable to reach to any point.... ...
3
votes
4answers
749 views

Sequence of solutions to $x\sin x=1$

Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed. Consider a sequence $x_n, n\ge1$ formed by positive solutions to ...
3
votes
2answers
2k views

Integrate $\csc^3{x} \ dx$

I found these step which explain how to integrate $\csc^3{x} \ dx$. I understand everything, except the step I highlighted below. How did we go from: $$\int\frac{\csc^2 x - \csc x \cot x}{\csc x - ...
1
vote
2answers
47 views

How to to a better approach for this :?

If, $$x\cos A+y\sin A=k=x\cos B+y\sin B$$ Then find $(\cos A)(\cos B)$, $(\sin A)(\sin B)$ and $\cos A+\cos B$ and express them in terms of $x,y,k$ I found a solution but it included a really ...
1
vote
2answers
82 views

Relationship among $A,B,C,D$ for $\cos A\cos B=\cos C\cos D$

While solving this Question, I could derive the following: As $\displaystyle 2\cos A\cos B=\cos(A-B)+\cos(A+B)$ substituting $A+B=90^\circ\iff B=90^\circ-A$ we get $\displaystyle 2\cos ...
8
votes
2answers
237 views

Compute $\int_0^1 \frac{\arcsin(x)}{x}dx$

$$\int_0^1 \frac{\arcsin(x)}{x}dx$$ This is a proposed for a Calculus II exam, and I have absolutely no idea how to solve it. Tried using Frullani or Lobachevsky integrals, or beta and gamma ...
6
votes
3answers
426 views

Prove trigonometry identity for $\cos A+\cos B+\cos C$

I humbly ask for help in the following problem. If \begin{equation} A+B+C=180 \end{equation} Then prove \begin{equation} \cos A+\cos B+\cos C=1+4\sin(A/2)\sin(B/2)\sin(C/2) \end{equation} How would ...
5
votes
1answer
587 views

The only two rational values for cosine and their connection to the Kummer Rings

I am trying to learn about Kummer Rings, and in particular what makes $n=3,4,6$ so special. (That is the Gaussian and Eisenstein integers) The only $\theta\in [0,\frac{\pi}{2}]$ which are rational ...