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

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152 views

Calculate vertical lines intersection hexagon at regular interval

I would like to calculate the total size of vertical lines that dissect an hexagon regular, like the one on the image. I would like to know the internal size of the blue lines inside the hexagon, ...
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49 views

From a light house $L$ two ships $P$ and $Q$ are observed in direction South West and $5^{\circ}$ East of South respectively. At same time $Q$…

Question : From a light house $L$ two ships $P$ and $Q$ are observed in direction South West and $5^{\circ}$ East of South respectively. At same time $Q$ is observed from point $P$ in South East ...
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82 views

How to solve sum of sines and cosines system of equations?

I have a set of equations to solve which in the following form: $ \cos(t_1 + t_2 + t_3 + t_4) + \sin(t_1 + t_2 - t_3 + t_4) + \cos(t_1 - t_4 + t_3 - t_5) + \sin(t_1 - t_2 + t_3 - t_5) + \cos(t_1 + ...
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87 views

Conditional inequality

Let x,y,z be positive reals with $xy+yz+zx=1$. Prove the inequality $$\sum_{cyc(x,y,z)}\frac {2x(1-x^2)}{(1+x^2)^2} \le \sum_{cyc(x,y,z)} \frac x{1+x^2}.$$ I substituted $x=tan\frac{\theta}2, ...
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27 views

Question about Chebyshev Polynomials in Beardon

I happen to be reading through Beardon's book, Iteration of Rational Functions, and I have come across a statement I don't quite believe. He uses it a little later on, so I'm concerned with clearing ...
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94 views

Proving a limit of a trigonometric function: $\lim_{x \to 2/\pi}\lfloor \sin \frac{1}{x} \rfloor=0$

$$\lim_{x \to \frac{2}{\pi}}\lfloor \sin \frac{1}{x} \rfloor=0$$ I need to prove the limit of this using the $\epsilon - \delta $ way but I don't know how to find $\delta$ when I'm given a ...
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119 views

Combinatorial proof for the formula of $\tan n\theta$

Is there any combinatorial proof of the formula for $\tan n\theta$ where $n\in \mathbb{N}$? Then proofs that I know are by Induction and using de Moivre's Formula but recently one of my friend asked ...
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107 views

What function has a 3D graph that will look like a spiral into a singularity?

I am trying to draw text spiraling into a black hole, from a more interesting slightly off-orthogonal viewpoint. I think a function that defines a black hole/singularity surface might look something ...
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80 views

Did Wolfram Alpha make this term up?

So I'm currently taking a Trigonometry class and I'm using the iOS version of the Wolfram-Alpha app to assist me with some of the more mundane calculations (Pythagorean Theorem, etc) and I've come ...
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146 views

Trigonometry of tetrahedron

I'm trying to develop the algebraic proofs for these two formulas that appear on the webpage below! The image below is of an unfolded non-regular tetrahedron. Triangle B represents the dihedral angle ...
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32 views

Parameterize the equation

Find a way of parameterizing the following curve: $y^2=\sin x $. I have already tried $x(t) = (\sqrt t, \sin^{-1} t) $ but this only gives part of the curve because of the nature of the sqrt function ...
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68 views

Expectation of product of cosines

I am reading a paper that starts with $$ E[ \cos( a(x-y) ] = E[ \cos(a x) \cos(a y) + \sin(a x) \sin(a y) ] $$ where the expectation is over $a$, then converts it into something of the form $$ = 2 ...
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240 views

Find the length of the longer diagonal on a trapezium with only 2 sides stated.

Im at a loss here, i know i have to divide the trapezium, but im still not sure which calculation is relevant to it then. Thanks in advance.
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68 views

Calculate angle of view from 2D image

I want to calculate the angle of view (or the field of view) from a photograph, without knowing anything about the camera, as to use that information in a 3D environment. I have to use trigonometry ...
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34 views

Formula for cos((2n+1)x) as polynomial of cos x

I am looking for a formula of cos((2n+1)x) that is polynomial of cos(x). For example, Is it known for any n?
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194 views

The geometry of a spiral made of adjacent right triangles

In the above figure (not sure if you can see it clearly or not), while using the old standard technique of plotting irrational numbers on number line, I saw this property. If we go on plotting ...
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122 views

Arctangents, Fibonacci numbers, and the golden ratio

In the course of doing scratchwork to answer this question, I had occasion to write the trigonometric identity $$ \arctan x- \arctan(1-x) = \arctan\left( \frac{1-2x}{x^2-x-1} \right). $$ Now notice ...
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206 views

Calculating point on sphere surface where sun reflection to a target point occurs

Imagine a mirror sphere at position O with radius R, and a target point at position P, at distance d from the sphere origin. There is an unknown point X on the surface of the sphere, where the light ...
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49 views

Ordinary Differential Equation with a trigonometric function: radius of convergence?

For the equation $$x^2y'' + y' + \tan(x)\,y = 0$$ establish lower bounds for the radius of convergence about the point $$x_0 = 1.$$
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100 views

How to find the period of a exponential function? $5\cdot(-1)^k$

Currently in class we are learning Euler's formula and how it relates to imaginary numbers, exponentials, and trig functions. I know the left hand of the equation is $e^{jwt}$ and I could easily use ...
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16 views

Nonnegativity on a special domain entails nonnegativity on the whole plane

Let $Q$ be a real bivariate polynomial such that $Q(x,\tan(x))\geq 0$ for any $x\not\in\{\pm\frac{\pi}{2}\}+(2\pi){\mathbb Z}$. Does it necessarily follow that $Q$ is nonnegative on the whole of ...
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65 views

Calculate area of this figure

I have an homework assignment where I have to calculate area of the figure underneath. I used the following formula to calculate the result $\frac {130 \cdot 55 \cdot sin35}{2} = 4188 m$ and then ...
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56 views

Would this thinking about the dot product hold?

Background today I completed the chapter on the dot product of vectors. But in trying to figure out exactly what the dot product is. I came to the conclusion that it can be interpreted as the length ...
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76 views

Using two chords and an angle to find center and radius of a circle

Hello, I am trying to solve the problem below. Is it possible to solve for the Center and Radius of the circle given the information provided, or is there something missing? I know how it's simple ...
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95 views

Proving $\frac\pi{22}\cos\frac\pi{22}+\frac{2\pi}{11}\cos\frac{5\pi }{22}+\frac{2\pi}{ 11}\cos\frac{9\pi}{22}+\frac\pi{22}\cos\frac{5\pi}{11}<\cdots$

$$(\frac{\pi}{22}) \cos (\frac{\pi}{22}) +(\frac{2\pi}{11}) \cos (\frac{5\pi }{22}) + (\frac{2\pi}{ 11}) \cos (\frac{9\pi}{22}) + (\frac{\pi}{22}) \cos(\frac{5\pi}{11}) < (\frac{\pi}{26}) ...
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117 views

Calculating rotated relative positions on 2D plane

For a game project, I need to calculate positions of items on a 2D plane relative to the camera. Camera can be rotated, and it's coordinates refer to it's center. In the attached images, ...
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145 views

Calculate a point on a geodesic line on an ellipsoid

I have a problem which i don't understand how to achieve. Maybe someone could sheed some light on it. Have a look at this picture: What I try to achieve is to determine the point D on the geodesic ...
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55 views

Is my answer correct, or textbook correct?

My answer is 9.45, textbook is 4.7 I did cos 38 = x / 12, 12 cos 38 = x 9.45 = x ty in advance
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71 views

Law of Sines Non-Euclidean geometry

Is the following Law of Sines valid on all surfaces isometric to a sphere? $$\frac{\sin A }{ \sin a }= \cdots = \frac{ \sin C }{ \sin c } = E.$$ And similarly, Is the following Law of hyperbolic ...
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93 views

Laplace transform of Differential Equation with a piecewise function

Hi I have this question and I am horribly stuck at one part and I cant seem to figure out if i did something wrong so any advice or help would be greatly apprecaited. Here is the question: ...
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48 views

find the angles of a given vector sum

Assume you have n vectors in 2D space, with different fixed magnitudes $l_i$. The problem is to find the angle of each vector such that vector sum is a specific vector. That is, $\sum l_i \cos ...
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35 views

$\frac {1 } {10 }(\sin(y_1+y_2)-\sin(x_1+x_2)+y_2-x_2)^2+(\cos(x_1+x_2)-\cos(y_1+y_2)+x_1-y_1)^2) \le (y_1-x_1)^2+(y_2-x_2)^2$?

Is it true that: $$\frac {1 } {10 }\left(\left(\sin(y_1+y_2)-\sin(x_1+x_2)+y_2-x_2\right)^2+\left(\cos(x_1+x_2)-\cos(y_1+y_2)+x_1-y_1\right)^2\right) \le (y_1-x_1)^2+(y_2-x_2)^2$$ I think I should ...
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203 views

Horizontal axis for reference angles

Why we always take the horizontal axis for reference angles? Is it by convention? Could it have been the y-axis? What advantages do we gain from taking the horizontal axis?
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32 views

Is this solution correct

Given $A+B=\frac{\pi}{4}$, find $(1+\tan A)(1+\tan B)$ My attempt: Since $\tan(A+B)=1=\frac{\tan(A)+\tan(B)}{1-\tan(A)\tan (B)}$, therefore $\tan(A)+\tan(B)+\tan(A)\tan(B)=1$, therefore, ...
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69 views

Relation between hyperbolic numbers and hyperbolic functions

Is there any relation between the hyperbolic (split-complex) numbers and hyperbolic trig functions? Or are they just named similarly by accident?
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76 views

Weighted sum of $\cos(nx)$ series

This is a follow up question to Prove $\frac{1}{2} + \cos(x) + \cos(2x) + \dots+ \cos(nx) = \frac{\sin(n+\frac{1}{2})x}{2\sin(\frac{1}{2}x)}$ for $x \neq 0, \pm 2\pi, \pm 4\pi,\dots$ I am looking ...
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48 views

Weird inequality answer, truncate or round?

When arriving at the final answer for a double inequality question, it appears that my text book has truncated one part and rounded the other. Is there some weird inequality rule that I don't know ...
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140 views

How to find Latitudes and Longitudes of projections of the vertices of a rectangular plane below earth's surface?

I want to find out the latitudes and longitudes of projections of the vertices of a rectangular plane inside the earth's surface. I know dimensions of rectangle, angles of orientation and latitude and ...
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73 views

Logarithm and “basic” functions.

To express the antiderivatives of $\frac{1}{x}$, we cannot apply the formula $\int x^n dx=\frac{x^{n+1}}{n+1}+C$ and we need to introduce a new function, the logarithm. But how can we prove that ...
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33 views

A Hard integral in 2-D.

I'm having a trouble integrating (in $\mathbb{R}^2$) the following formula: $$\frac{t}{|B(x,t)|}\int_{B(x,t)} \frac{||y||}{(t-||x-y||^2)^{\frac{1}{2}}} dy $$ where $B(x,t)$ is the ball with center ...
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101 views

Solving an equation with $\arccos(x)$ and $\sin(\arccos(x))$

I want to solve this equation, determining y (all others letters are constants) : $$2 \arccos(3+(1.6y-80)/R) - \sin(2\arccos(3+(1.6y-80)/R)) = 2π(1-P)$$ I've try to use some automatic solvers but ...
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34 views

Given only this graph of a certain cosine function, how can I determine what it's “c” is? (click to see graph)

Although my answer key says $c$ is supposedly $\frac{2\pi}{3}$, I actually get $c = \frac{4\pi}{3}$ Thus, $y$ should equal $\frac{5}{2}cos(\frac{\pi(x)}{2}-\frac{4\pi}{3})-0.5$ , right?
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44 views

What is the number of x-intercepts in this graph of sine?

The function : $y=3-4\sin(2\pi x-3\pi)$ .. how many $x$-intercepts over the interval $[0,2]$? I am confused if they're 3 or 5 because there are 3 $x$-intercepts that are really intercepting ...
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198 views

arrange div elements in circle and square

I n number of divs which are arranged in a circle using javascript. Right now i set the dimension of each div to 40*40. Below is what i am able to achieve so far. This is how i find X & Y of each ...
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100 views

proving $\tan^{-1}(x)+\tan^{-1}(y) = \pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right)\;,$ when $x>0,y>0,xy>1$

(1) How ca we prove $\displaystyle \tan^{-1}(x)+\tan^{-1}(y) = \pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right)\;,$ when $x>0,y>0,xy>1$ (2) How ca we prove $\displaystyle \tan^{-1}(x)+\tan^{-1}(y) ...
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57 views

How to I solve the inverse $(e^x – je^x)/(e^x+e^x)$?

I have tried using this method https://www.youtube.com/watch?v=V-LJWfuoCDs. But I am getting zero on one side thus cancelling $e^x$, which means that the the answer I get will not be in form of ln. Is ...
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54 views

Relation between $\sin(t)$($\cos(t)$) and $\sin(at)$ ($\cos(at)$) when both are rational

This question relates to Parametric equations where sin(t) and cos(t) must be rational. Suppose it is given that $\cos(t)$ and $\sin(t)$ are both rational and also $\cos(at)$ and $\sin(at)$, where ...
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47 views

$\cos(2\arccos(\frac{a}{a+1})x$

I have trying to prove that this cosine map: $$\frac{r}{4}((a+1)\cos\left(2\arccos\left(\frac{a}{a+1}\right)\ \left(X_n-\frac12\right)-a\right)$$ is a logistic map. What I have done so far: Using ...
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46 views

A right triangle with sides

Imagine a right triangle with sides: Long side C is $4n$, sides $b$ and $a$ are $2n$ and $n$, where $n$ is an integer. How many right triangles are of this form? My attempt: $$16n^2 = 4n^2 + n^2$$ ...
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116 views

Formula for nth derivative of $\arcsin^k(x/2)$

I need to find formula for $n$-th derivative of $\arcsin^k(\frac{x}{2})$. I have found formula for ...