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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186 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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179 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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135 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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124 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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187 views

Integral of a gaussian function of trigonometric functions

I need help with the analytical solution of this integral: ...
6
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136 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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567 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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99 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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50 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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144 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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96 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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112 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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71 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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163 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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68 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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114 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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78 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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179 views

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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301 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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223 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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289 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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338 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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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 ...
3
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94 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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100 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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70 views

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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143 views

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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102 views

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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63 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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60 views

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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302 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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991 views

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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208 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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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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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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104 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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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 ...
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100 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 ...
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41 views

How find the range value $a^2+b^2$ if $\cos{(a\sin{x})}=\sin{(b\cos{x})}$ have no solution

if the equation $$\cos{(a\sin{x})}=\sin{(b\cos{x})}$$ have no zero solution,then $a^2+b^2$ range of value $A:[0,\dfrac{\pi}{4})$,$B: [0,\dfrac{\pi^2}{2})$,$C: ...
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Alternative Proof for “Roots of Mertens Function-Farey Sequence-Cosines Relations”

You can write Merten's function as $$ M(n)= \sum_{a\in \mathcal{F}_n} e^{2\pi i a} , $$ where $\mathcal{F}_n$ is the Farey sequence of order $n$. The sum may be split into imaginary and real ...
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45 views

Integer Factorization via Trigonometry

Nearly 20 years ago, I was sitting in a physics class in high school when a "dumb" question occurred to me: If two pendulums with unknown (different) frequencies started oscillating at the same time ...
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59 views

Bretschneider-Brahmagupta-Heron Proof

Derive Bretschneider's formula, Brahmagupta's formula and Heron's formula in one memorable elegant proof. I ask this question merely to see the creativity of the MSE community when it comes to ...
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14 views

Taking into account Camera Direction in 3d models using Trig

I have been working on a 3d rendering project as code. I am a bit stumped on the math though. I know how to render points using arctangent on an HTML canvas using JavaScript, like this: ...
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46 views

Trigonometric functions of angle fractions

I've just encountered a problem that seems to me interesting enough so that some result exists on the subject. I was working on a problem in complex analysis, in which I needed the fifth root of a ...
2
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50 views

Pipe-fitting problem 3D

I have a 3D pipe-fitting problem for which I was able to write the following equations: $$ y = \tan (a)\sqrt{x^2 + z^2}\\ z = \tan (b)\sqrt{x^2 + y^2}\\ y = \sin (a)\sqrt{x^2 + y^2 + z^2}\\ z = \sin ...
2
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65 views

Radius of circle by knowing a cross section.

I have a curve on an ellipse where I know the length of a cross section and need to find out it's radius (vertically and horizontally) and calculate the angle of the curve. In the following diagram ...
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96 views

Trigonometry question (acute angle triangle)

Let $ABC$ be an acute angle triangle. Show that : \begin{equation} \sum _{cyc}(\sin2B+\sin2C)^{2} \sin A \leq 12 \sin A \sin B \sin C \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(*)\end{equation} ...
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57 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 ...