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

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On Ramanujan's Question 359

In JIMS 4, p.78, Question 359 was asked by Ramanujan. If, $\sin(x+y) = 2\sin\big(\tfrac{1}{2}(x-y)\big)\tag1$ $\sin(y+z) = 2\sin\big(\tfrac{1}{2}(y-z)\big)\tag2$ prove that, $\big(\tfrac{1}{2}\sin ...
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234 views

What comes after $\cos(\tfrac{2\pi}{7})^{1/3}+\cos(\tfrac{4\pi}{7})^{1/3}+\cos(\tfrac{6\pi}{7})^{1/3}$?

We have, $$\big(\cos(\tfrac{2\pi}{5})^{1/2}+(-\cos(\tfrac{4\pi}{5}))^{1/2}\big)^2 = \tfrac{1}{2}\left(\tfrac{-1+\sqrt{5}}{2}\right)^3\tag{1}$$ ...
9
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182 views

How find this value $\sum_{i_1=1}^n \sum_{i_2=1}^n \cdots\sum_{i_k=1}^n \cos\frac{k(i^k_1+i^k_2+\cdots+i^k_k)}{n}$

Question: Find this value $$\sum_{i_{1} = 1}^{n}\sum_{i_{2} = 1}^{n}\cdots\sum_{i_{k} = 1}^{n} \cos\left(k\left[\vphantom{\Large A}\, i_{1}^{k} + i_{2}^{k} + \cdots +i_{k}^{k}\,\right] \over ...
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139 views

Verifying the inequality $\sum_{i=1}^n \frac{\cos y_i}{\sin x_i}≤ \sum_{i=1}^n\cot x_i $

Let $\sum_{i=1}^n x_i=\sum_{i=1}^ny_i= \pi$ , where $n>1$ and $x_i >0 , y_i>0 , \forall i=1,2,..,n$. How can we prove that $$\sum_{i=1}^n \frac{\cos y_i}{\sin x_i}≤ \sum_{i=1}^n\cot x_i $$ ? ...
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175 views

Problem : If $\sin^2\theta = \frac{x^2+y^2+1}{2x}$ , $x$ must be …

Problem : If $\sin^2\theta = \frac{x^2+y^2+1}{2x}$ , $x$ must be (a) $1$ (b) $-2$ (c) $-3$ (d) $ 2$ My approach : Since $0 \leq \sin^2\theta \leq 1$ $\Rightarrow 0 \leq ...
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107 views

What transcendental numbers are produced by $\sin{\alpha}$ when $\alpha$ is algebraic/constructible/rational (in radians)?

I know that by Lindemann–Weierstrass theorem(LW) sine and cosine of non-zero algebraic numbers (in radians) produce results that are transcendental. My question is what are the transcendentals ...
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201 views

Integral of a gaussian function of trigonometric functions

I need help with the analytical solution of this integral: ...
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145 views

Rational multiples of $\pi/2$ whose sines are also rational

Let $f(x)=\sin(x\frac{\pi}{2})$. Let $R$ the set of $x$ such that : $0\le x\le 1$ $x \in \mathbb Q$ $f(x) \in \mathbb Q$ Hence, $0\in R$ as $f(0)=0$. $1\in R$ as f(1)=1. And $\frac{1}{3}\in R$ as ...
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582 views

Nasty Integral - Closed form solution?

Any suggestions on how to integrate this beast?: $$\int_0^{\omega_t}\int_{\omega_t}^f\sin^2(\omega_{12}/2)\sin^2(\omega_{23}/2)d\omega_{23}d\omega_{12}$$ where: $f = 2\pi+2\tan^{-1}(y,x)$ $y = ...
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101 views

is the unique solution of $\cos t = t$ a transcendental number?

let $\alpha$ be the unique fixed point of $\cos:\mathbb{R} \rightarrow [-1,1]$ for any $t \in \mathbb{R} \setminus\{0\}$ if $t$ is algebraic then $\cos t$ is transcendental. thus if $\alpha$ were ...
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143 views

Obtaining a deeper understanding of lower level Mathematics

I am a college student, at a community college and I am in the process of obtaining an associates degree in general science with a specialization in mathematics in hope of transferring to a university ...
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56 views

Sum of squares of cotangents (Check properly of expression)

I found exercise in "Introduction to algebra" Part I (A.I. Kostrikin) Check expression $\sum_{k=1}^n\cot^2\frac{k\pi}{2n+1}=\frac{n(2n-1)}{3}$ for $n=1,2,3,4,5$. For $n=1,2$ it is simple. ...
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155 views

Determine the general solution for $2\cos^2x-5\cos x+2=0$

Determine the general solution for $ 2\cos^{2}x-5\cos x+2=0$ My attempt: $2u^2 - 5u + 2 = 0$ $(2u - 1)(u - 2) = 0$ $u = \frac{1}{2}$ or $u = 2$ $\cos(x) = \frac{1}{2}$ or $\cos(x) = 2$ The ...
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69 views

A trigonometric integral with sin(cos(x)) in exponent

Evaluate: $$\int_0^{\pi} x\csc^{\sin(\cos x)}(x)\,dx$$ I honestly don't know how to deal with this case. If I apply the property $\int_a^b f(x)\,dx=\int_a^b f(a+b-x)\,dx$, I get: $$\int_0^{\pi} ...
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46 views

Maximum and Minimum value of an inverse function

Find the maximum and minimum value of $\arcsin \left(x\right)^3+\arccos \left(x\right)^3$. given that $-1\le x\le 1$ I have solved the problem but i am just curious to know if there are ...
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110 views

Proof that cos(1) is transcendental?

So, I was playing around on Wolfram|Alpha (as we nerds like to do) and it said cos(1) was transcendental. Could someone provide me with the proof that cos(1) is transcendental?
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Trigonometry or inequality problem

Today, I saw this question: If $x,y,z \in [0,\frac\pi 2]$, $x+y+z=\frac{3\pi}{4}$ and $\sec^2(x)\sec^2(y)\sec^2(z)=8$, calculate $E=\tan x\tan y+\tan y\tan z+\tan z\tan x$ My first thought was ...
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82 views

How prove this inequality $\sin^2{\frac{A}{2}}+\sin^3{\frac{B}{2}}+\sin^4{\frac{C}{2}}\ge\frac{7}{16}$

in $\Delta ABC$,such $$5\cos{A}+6\cos{B}+7\cos{C}=9$$ show that $$\sin^2{\dfrac{A}{2}}+\sin^3{\dfrac{B}{2}}+\sin^4{\dfrac{C}{2}}\ge\dfrac{7}{16}$$ By the way: This inequality is my favourite ...
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202 views

Geometric interpretation of hyperbolic functions

When proving identities like $$\cosh(2x)=\cosh^2(x)+\sinh^2(x)$$ $$\cosh^2(x)=\sinh^2(x)+1$$ algebraically, I am beset by the feeling that there should be a geometrical interpretation that makes them ...
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73 views

Conditions to satisfy trigonometric inequality

I'm looking for sufficient (and necessary would be good too) conditions on $a,b,c$ such that \begin{align} a\cos\phi + b \cos 3\phi + c \cos 5\phi \geq -1 \hspace{20pt} (\forall \phi) \end{align} ...
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115 views

Prove That $1/\sin(i \pi/n)+1/\sin(j \pi/n) \ne 2$ When $n$ is Odd

The original problem is: Prove that $1/\sin(i \pi/n)+1/\sin(j \pi/n) \ne 2$ when $n$ is odd. I tried, and found that although everything looks similar, I actually know nothing about this kind of ...
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88 views

Generalized sine integral $ \int_0^\infty \sin^m(x^n)/x^p\,\mathrm dx$

I have seen that both the integrals $$ \color{black}{ \int_0^\infty \operatorname{sinc}(x^n)\,\mathrm{d}x = \frac{1}{n-1} \cos\left( \frac{\pi}{2n}\right)\Gamma ...
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How to find the maximum diagonal length inside a dodecahedron

I am trying to find the maximum length of a diagonal inside a dodecahedron with a side length of 2.319914107*10^89 meters. I am not sure if any other information than that is needed, if it is I ...
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341 views

Calculating equidistant points around an ellipse arc

As an extension to this question on equiangular fisheye distortion, how can I calculate equidistant points around an ellipse (or 1/4 segment of) given it's aspect ratio? When it's circular, I can use ...
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237 views

Reprojecting/converting an orthographic image/grid into a cartesian grid

I'm trying to dewarp a fisheye image into a simple rectilinear image of a subset of the fisheye. As part of this, I'm trying to map the azimuth/altitude values into a point on the image. The points ...
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309 views

$\arctan$ of constant multiplied $\tan \varphi$

Helllo, I'm new here. I will try this great place to get my answer. I have a next problem: $$\vartheta = \arctan(K \tan \varphi)$$ $K > 0$ Where $K$ is a constant. If $K = 1$ then $\vartheta = ...
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366 views

Find roots of sum of sinusoids

Given this function and an initial point, find the next root: $$ \begin{align} f(t) & = -L\\ & {} + A \sin(\Theta_1 + \omega_1 t) \\ & {} +B \cos(\Theta_1 + \omega_1 t)\\ & {} - ...
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94 views

how understand if a segment is inside a lissajous curve

i am a programmer and not a math guru, but i like geometry. so if i'm not accurate in math terminology or i have folly question please sorry me. i'm drawing with a programming language the lissajous ...
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How to solve this problem 4

Question $1$: Is $\frac{1}{\pi}\arccos\left(\frac{{\sqrt{2*\sqrt{2*\sqrt{2}*...n}}}}{2}\right)$ always a rational number when each$*$ is either $+$ or $-$ and $n$ may or may not be infinite? ...
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49 views

Link between a cubic polynomial and a trig identity

Alright, so I am told to prove that: $$\tan (3A) = \frac{3\tan(A)-\tan^3(A)}{1-3\tan^2(A)}$$ This can be pretty easily done by applying the $\tan$ addition formula, taking the angles $2A$ and $A$, ...
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32 views

unit circle trigonometry where angles is greater than 90

how is possible to have sin of angle greater than 90. if sin is ratio of opposite side and hypotenuse in right angle triangle then triangle with one of the angle greater than 90 can not be right angle ...
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158 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)}= ...
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107 views

$\sin n$ having a closed form expression with no complex terms?

[All angles are in degrees] I have heard that we cannot express $\sin 1$ in closed form with no complex terms. However, I know that we can derive $\sin 18$ by solving $\cos 3x = \sin 2x$. Thus we can ...
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63 views

When $\cos{n^{\circ}}$ can be expressed in real radicals?

So, the question is: $\cos{n^{\circ}}$ can be expressed in real radicals iff $3 \mid n$? Is it true? The first part is easy: if $3 \mid n$ we can express it, because ...
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Partial fraction development of $\cot \pi z$

"Compute the values $\sum_{n=1}^\infty \dfrac{1}{n^2}$ and $\sum_{n=1}^\infty \dfrac{1}{n^4}$ by comparison to the partial fraction development of $\cot \pi z$." I'm not sure what "the partial ...
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Find all $\theta$ such that sin$\theta$ and cos$\theta$ are both rational number.

Find all $\theta$ such that sin$\theta$ and cos$\theta$ are both rational number. I thought this question might have been asked by someone else, but I couldn't find any. Currently I'm studying ...
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65 views

Proof of a series containing normal as well as hyperbolic functions

Show that $$\sum _{n=0}^{\infty}\frac{\sinh(\pi n \sqrt 2)- \sin(\pi n \sqrt 2)}{n^3{\cosh(\pi n \sqrt 2})- \cos( \pi n \sqrt 2)}=\frac{ \pi^3}{18 \sqrt 2}$$ I have no hint as to how to even start.
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Volume of spheroidal cap

A spheroid is bisected into two spheroidal caps by a plane, such that the shape of the area of the plane inside the spheroid is elliptical. The alignment of the plane is defined by two angles theta1 ...
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309 views

Trigonometric inequality proof

Can anyone help me in proving that $$\cos\theta > \frac{\left(x^a\cos\theta-(x-1\right)^a\cos\frac{\ln x\theta}{\ln(x-1)})\cos(\theta+\gamma)}{\cos\gamma},$$ where $a<1$, $x\in \mathbb{N}$, and ...
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Proof of the sine rule

So I made my first attempt at a proof. I think it turned out well. Maybe not. But I was wondering if someone could take a look at it and tell me what they think. I'd be glad to hear some criticism on ...
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215 views

Product of sines

I am looking to evaluate $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n}$$ without using complex numbers. I can show the result if $n$ is a power of $2$, but if $n$ is anything else I reach a point where I ...
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49 views

secants, exponentials, quotient structures

Trigonometric functions are . . . . . . somewhat like exponential functions. If $f$ is an exponential function, then $\displaystyle \frac{\prod_i ...
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103 views

Functional inverse of $(a + b\sin\theta)^2\tan\theta$

So, I revisited to the situation involved in my previous question, with the intent of generalizing it to any two masses and charges. When I started going through the model again, beginning with the ...
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106 views

Solving a linear trigonometric equation

Let $n$ be a natural number. For $a_i,\omega_i,\varphi_i \in \mathbb{R}$ how can one find solutions $x \in \mathbb{R}$ for the equation: $$\sum_{i=1}^n a_i \cos( \omega_i \cdot (x-\varphi_i)) = 0$$
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196 views

Maximum size of a rotated-then-cropped rectangle

With regard to topic/question New size of a rotated-then-cropped rectangle: The answer by Isaac, the maximum area is $b^2\csc\alpha\sec\alpha$ when $x=0.5b\csc\alpha = 0.5b/\sin\alpha$ seems to ...
3
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0answers
105 views

How do I calculate the size of padded envelope needed for a given size box?

When you put a box of dimensions (W x H X D) in a padded envelope of dimensions (X x Y), what is the mathematics? The padded envelope also has to have a flap of length (F), it also has welded seams ...
2
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97 views

Alternative to sin and cos

I was reading something on the Internet the other day, and I swear I came across a reference to an alternative sine function [which I now cannot find any mention of]. The usual sine function starts ...
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56 views

To find a fifth degree equation by using circles and lines that cannot be solved by radicals

An example quintic whose roots cannot be expressed by radicals is $x^5 - x + 1 = 0$. I asked a geometry question about a fifth degree equation long time ago . I had an equation in the question. It ...
2
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49 views

Angular velocity of the minute hand

The exercise is to calculate the angular velocity (in radians per hour) of the rotation of: the hour hand, and the minute hand (of the clock). Neither of my answers coincides with the answers in ...