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

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4
votes
2answers
465 views

Fixed-Point Iteration method unable to converge to any of a function's infinte roots

An equation is given to me which has to be solved by direct iteration method: $$\sin(x) = {x+1 \over x-1}$$ or $$f(x)=\sin(x)-{x+1 \over x-1} = 0$$ I follow the following procedure with reasons ...
2
votes
1answer
28 views

How would one evaluate $\sin(72\pi/11)$?

How would one evaluate $\sin(\frac {72\pi} {11})$?. The prime number in the bottom is getting me stuck. I couldn't see how to use it using the sum of two angles trig identity.
1
vote
1answer
47 views

How to compute $\lim\limits_{x \rightarrow 0} \frac{1}{x^2}\int_0^{G(x)} \arctan(s+2s^2) ds$

Suppose $g$ is a function that has its derivatives everywhere and $G(x)=\int_0^x g(t)dt$. How to compute $\lim\limits_{x \rightarrow 0} \frac{1}{x^2}\int_0^{G(x)} \arctan(s+2s^2) ds$? To start ...
0
votes
1answer
24 views

Trigonometry markup

Imagine we have the following problem; $$\cos(x) = \cos(a) \Rightarrow x=a+k\times 2\pi\\ or \\x=-a+k\times 2\pi$$ And we have the following answers.. : $$a=\frac{\pi}{3} \\or \\a=-\frac{\pi}{3}$$ ...
0
votes
1answer
20 views

Trigonometry Proving

If $\sin\theta + \sin\alpha=x$ and $\cos\theta + \cos\alpha=y$, prove that ; $$\frac{\tan(\theta - \alpha)}{2} = \pm\sqrt{\frac{4-x^2-y^2}{x^2+y^2}}$$ Attempts: Here $\sin\theta + \sin\alpha=x$ ...
0
votes
1answer
41 views

Finding if there is a maximum or minimum on a curve?

My apologies for being very brief with this question, the reason for this is because I don't know where to start. The question is as follows: A curve has the equation $\lambda \cosh(x) + \sinh(x)$, ...
19
votes
0answers
775 views

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, ...
9
votes
0answers
222 views

How to 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 ...
7
votes
0answers
286 views

Integral of a gaussian function of trigonometric functions

I need help with the analytical solution of this integral: ...
6
votes
0answers
251 views

Maximal minimum for a sum of two (or more) cosines

Please prove (or disprove, and give the correct answer): $$2 =\mathrm{argmax}_{r\geq 1}\min_{x\in \mathbb{R}}\left[\cos\left(x\right)+\cos\left(rx\right)\right] $$ In other words, find $r \geq 1$, ...
6
votes
0answers
174 views

The minimum of $I_{n,k}=\int_0^{2\pi}\sqrt{3+2\cos(nx)+2\cos(kx)+2\cos(nx+kx)}dx$ is attained for $k=n$

I have the following conjecture: ``For each given $n\in\mathbb{N},\ n\ge 2$ the minimum of the sequence of integrals $I_{n,k}=\int_0^{2\pi}\sqrt{3+2\cos(nx)+2\cos(kx)+2\cos(nx+kx)}dx,\ k=1,2,\dots,n$ ...
6
votes
0answers
114 views

Condition for trigonometric inequality

I want to prove the following statement: Suppose $\frac{1}{4}(\cos(\theta_1)+\cos(\theta_2))^2+\lambda^2(a\sin(\theta_1)+b\sin(\theta_2))^2\leq 1$ holds for all $\theta_1,\theta_2\in[-\pi,\pi]$, then ...
6
votes
0answers
231 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 ...
6
votes
0answers
658 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 = ...
5
votes
0answers
67 views

How do I evaluate this integral :$ \int_0^\frac{\pi}{2} [\text{chi}(\cot^2x)+\text{shi}(\cot^2x)]\csc^2(x)e^{-\csc^2(x)}dx$?

I have tried to evaluate this integral :$$ \int_0^\frac{\pi}{2} [\text{chi}(\cot^2x)+\text{shi}(\cot^2 x)]\csc^2(x)e^{-\csc^2(x)}dx $$ where, $\text{shi}(x)=\int_{0}^{x}\frac{\sinh t}{t}dt$ , ...
5
votes
0answers
46 views

Inverting a somewhat complicated trig transformation

I have, for somewhat arcane reasons, the following transformation between given spherical angles $(\theta,\phi)$ (with $-\pi/2 < \theta < \pi/2$) and new angles $(\theta',\phi')$ (with $-\pi/2 ...
5
votes
0answers
48 views

Generalized Trigonometric Functions in terms of exponentials and roots of unity

I am trying to come up with generalized trigonometric functions using the exponential definition that we use today for the trig functions sine and cosine $$\sin x=\frac{e^{ix}-e^{-ix}}{2i}; \cos x ...
5
votes
0answers
157 views

resizing rectangle within triangle

Imagine I have a parking lot that changes in width and length and in number of levels, and all of the levels need to be visible to a cctv camera at a fixed position, and I would want the camera to see ...
5
votes
0answers
124 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} ...
5
votes
0answers
463 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 ...
5
votes
0answers
134 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 ...
5
votes
0answers
115 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 ...
4
votes
0answers
85 views

Trigonometric Expression for $1 + \cos \alpha + \cos 2\alpha + \cdots + \cos n \alpha$ using complex numbers

This question is not a duplicate because I am asked here to use the fact that $1 + \cos \alpha + \cos 2 \alpha + \cdots + \cos n \alpha = Re (1 + z + z^{2} + \cdots + z^{n})$, where the question this ...
4
votes
0answers
31 views

Get bounding rectangle segments of a rotated rectangle (matrix?)

My problem: I have: $x$, $y$ & $\alpha$ - the aspect ratio $o$:$p$ (red rectangle) I want to have $n$ & $m$ in dependancy of $x, y, \alpha, o, p$ I tried to figure it out with ...
4
votes
0answers
158 views

Combinatorics and trigonometry identity

Prove the following: $\displaystyle\prod_{n=1}^{180}\left(\cos{\left(\dfrac{n\pi}{180}\right)}+2\right)=\displaystyle\sum_{n=0}^{89}\binom{180}{2n+1}\left(\dfrac{3}{4}\right)^n$. We can state this ...
4
votes
0answers
68 views

Why is the infinite product of this quotient of $\sin$'s equal to $\left(\frac{3}{\pi}\right)^{2}$[SOLVED]

I was intrigued by this answer the other day, but perhaps lack a little bit of the necessary background to understand a certain step. Namely, the fact that $$\prod_{n=1}^{+\infty} ...
4
votes
0answers
40 views

Use Trigonometry to make a Marksman's Sight

In the Civil War, marksmen used something called a "stadia" which is nothing more than a piece of brass with a triangular hole cut out and a slide with marks of varying distances. Line your target up ...
4
votes
0answers
54 views

Rational distances from the corners of a unit square

Is there a point all of whose distances from the corners of the unit square are rational?
4
votes
0answers
30 views

Iterated circumcenters - proving collinearity and establishing distance ratios

Let $P_0, P_1, P_2$ be three points on the circumference of a circle with radius $1$, where $P_1P_2 = t < 2$. For each $i \ge 3$, define $P_i$ to be the centre of the circumcircle of $\triangle ...
4
votes
0answers
61 views

$\int_{-\pi/2}^{\pi/2} \cos(a \cos\theta) e^{im\theta} e^{-ib\sin\theta} \mathrm{d}\theta $ Integration

I am struggling to find the integration of the expression below, $$\int_{-\pi/2}^{\pi/2} \cos(a \cos\theta) e^{im\theta} e^{-ib\sin\theta} \mathrm{d}\theta $$ where $a$ and $b$ are arbitrary constant ...
4
votes
0answers
89 views

How prove $\root 4\of{\frac{1}{2}\sin x\cos z}+\root 4\of{\frac{1}{2}\cos x\sin z}=\root{12}\of{\sin 2y} $?

Let $\sin(x+y) = 2\sin\left(\dfrac{x-y}{2}\right)$ and $\sin(y+z) = 2\sin\left(\dfrac{y-z}{2}\right)$. How prove $\root 4\of{\frac{1}{2}\sin x\cos z}+\root 4\of{\frac{1}{2}\cos x\sin ...
4
votes
0answers
109 views

Graphs of interesting integrals of the form: $\int \sin^a(x^a)\cos^a(x^a)$

Here are a few graphs of the form:- $$\int \sin^a(x^a)\cos^a(x^a)dx$$ Where $a$ is an even, positive integer. $a = 2$ $a = 4$ $a = 6$ Now, a few graphs of the form:- $$\int ...
4
votes
0answers
103 views

How to solve this equation in $ \mathbb{C} $?

From a small simple calculation , we get the following formulas: $ \begin{cases} e^x = \Big( \displaystyle \sum_{n \geq 0} \frac{x^{3n}}{(3n)!} \Big) + \Big( \displaystyle \sum_{n \geq 0} ...
4
votes
0answers
101 views

More on primes $p=u^2+27v^2$ and roots of unity

Given, $$p=u^2+27v^2=3m+1\tag1$$ and the cubic, $$x^3+x^2-mx+N=0\tag2$$ with its constant expressed in terms of $(1)$ as, $$N = \frac{1}{27}(1-3p\pm2pu)\tag3$$ and the sign $\pm u$ chosen ...
4
votes
0answers
105 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} ...
4
votes
0answers
128 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 ...
4
votes
0answers
350 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 ...
4
votes
0answers
507 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 = ...
4
votes
0answers
533 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)\\ & {} - ...
4
votes
0answers
106 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 ...
3
votes
0answers
29 views

Is it always possible to find the roots of $P(z)=az^4+bz^3+cz^2+bz+a$, where $a,b,c \in \mathbb{R}^*$, by first dividing both sides by $z^2$?

A classic way to solve quartics in the form $P(z)=az^4+bz^3+cz^2+bz+a$, if we know that the roots lie on the unit circle, is to divide both sides by $z^2$ and then use the fact that if $$z=\cos \theta ...
3
votes
0answers
58 views

PDF of Random Variable $\sin\alpha \cdot \cos\beta$ with $\alpha,\beta$ uniform

As part of a bigger problem, I want to compute the probability density $f_Z(z)$ of $$Z = \sin\alpha \cdot \cos\beta$$ where $\alpha, \beta$ are random variables, independently and uniformly ...
2
votes
0answers
34 views

Expanding trigonometric functions with binomial expansion

I was challenged to take $\cos^{\pi}(\pi)$ and expand it using binomial expansion and $\cos(x)=\frac{e^{xi}+e^{-xi}}2$, which I tried: $$\cos^{\pi}(\pi)=\left(\frac{e^{\pi i}+e^{-\pi ...
2
votes
0answers
34 views

Find elevator height given rope length?

This question is deceptively difficult. I feel like it's probably some classic example somewhere, but I'm not sure how to describe it in enough detail to get valid results in searching online. ...
1
vote
0answers
40 views

Find $\sum_{i=1}^{2000}\gcd(i,2000)\cos\left(\frac{2\pi\ i}{2000}\right)$

What is the value of the following sum? $$\sum_{i=1}^{2000}\gcd(i,2000)\cos\left(\frac{2\pi\ i}{2000}\right)$$ where $\gcd$ is the greatest common divisor.
0
votes
0answers
7 views

Difficult problem involving a percentage of the period of a sinusoid

Im having difficulty intuitively understanding how to solve this problem: $x(t) = A*cos(\Omega*t + \phi)$ $A > 0$ $\phi$ range is $(−\pi,\pi]$. $x(t) ≥ 2.4$ for $18$% of each period takes ...
0
votes
0answers
32 views

How can I change a summation of cosines to a product of cosines for higher degree functions?

I was wondering if I can get an alternate form of a sum over cosines $$\sum_{n=1}^m\cos{f(n)}$$ and I found that I can. We must however make a modification to the upper limit with $m\rightarrow2^m$. ...
0
votes
0answers
25 views

Find the rotation angles of a 2-D rotation matrix between two vectors

I am trying to solve the following to find $\theta$. I was given two vectors $\begin{bmatrix}-4.95 \\ -.7\end{bmatrix}$ and $\begin{bmatrix}3 \\ 4 \end{bmatrix}$ and asked to compute the rotation ...
0
votes
0answers
23 views

Find the third angle

Three planes are orthogonal to each other. I have found the rotation about the 2 axis (x and y). Is there a way to find the third angle around z provided the angles around x(70 degrees) and y(-1 ...
-1
votes
0answers
25 views

parametr with trigonometric equations

can you help me with this equation: $$ \frac{4p+3}{6} - \frac{\sin{8x}}{2} - (p+\frac{2}{3})\sin(4x-\frac{\pi}{4})=0 $$ for which values of $p$ equation has has 3 distinct solutions in range ...