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

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56 views

Book on higher-power trigonometric equation simplification techniques

Recently, I have become fascinated about learning techniques for simplifying experimentally derived trigonometric mathematical models that are raised to higher powers. Are there any good references ...
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92 views

Inequality sine power series

How can we show, for $k\geq 1$ and $x \geq 0$, the inequality below by induction? $\displaystyle \sin x \geq \sum_{n=1}^{2k} (-1)^{n+1} \frac 1{ (2n - 1)! }x^{2n-1} $ The base case $k = 1$ gives ...
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103 views

conjecture regarding the cosine fixed point

context/motivation if the angle on a calculator is set to radians, then it is very easy to demonstrate that iteration of $cos x$ (for arbitrary initial x) converges - simply keep pressing the ...
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74 views

Derivative of trig function

Find the second derivative of $ \arcsin(2x^3) $ The solution says for the first derivative : $ \dfrac{1}{\sqrt{1-(2x^3)^2}} \cdot 6x^2 = \dfrac{6x^2}{\sqrt{1-4x^6}} $ When i answered the first ...
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455 views

Reduction formula for a trigonometric integral

I have come upon the following trigonometric integral: $$\int (\alpha + \sin x)^n \cos^2 x\,\mathrm{d}x,$$ where $\alpha \in \mathbb{R}$ is an arbitrary real constant and $n \in \mathbb{N}$ is a ...
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86 views

Relating $x$ to $\frac{dx}{dt}$ in a right triangle.

I have a right triangle with sides of $x$ and $y$. I know $y$ is a constant (500) and that $\frac{d \theta}{dt}$ (where $\theta$ is the angle opposite from side $x$) is also constant ($8\pi$ rad/s). I ...
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58 views

Reference request: Another tangent half-angle formula

Wikipedia's "Tangent half-angle formula" article lists these: \begin{align} \tan\frac\theta2 & = \frac{\sin\theta}{1+\cos\theta} = \frac{1-\cos\theta}{\sin\theta} = \frac{\tan\theta}{1 + ...
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121 views

What's the acceptance of rational trigonometry in current mathematics courses?

I've been reading about Wildberger's rational trigonometry and I'm willing to learn it. I'm wondering if it's usage is accepted in undergraduate mathematics courses. It seems there's a redefinition on ...
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105 views

Trigonometric functions of rational fractions of pi

Consider rational numbers $\frac{m}{n}$ and $\frac{m'}{n'}$, where $0<\frac{m}{n}, \frac{m'}{n'} <1$. Then $$\sin^2 (\tfrac{m}{n} \pi) = 2 \sin^2 (\tfrac{m'}{n'} \pi)$$ When $\frac{m}{n} = ...
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256 views

Position of a point on a moving unit circle?

Suppose a unit circle is moving in a horizontal line at a speed given by $f(t)$ , and a point on the unit circle is moving counterclockwise at a speed given by $g(t)$ , and the initial position of the ...
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101 views

Bernoulli generating function and cotangent

May I ask for a little help in solving a problem about Bernoulli number generating function? Bernoulli number generating function is given by: $$f(z):=\begin{cases} \frac{z}{e^{z}-1} & z \in ...
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114 views

At large times, $\sin(\omega t)$ tends to zero?

While doing a calculation in quantum mechanics, I got a expression $\sin(\omega t)$, and my prof said if I consider the consider at large times, then i can assume that this goes to zero because at ...
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641 views

Solving the equation $\cos(x) \cdot \cosh (x) + 1 = 0$

$$\cos(x) \cdot \cosh (x) + 1 = 0$$ Sorry I am a software developer and I have forgotten this part of mathematics! What is the value of $x$ in the above equation? I need the steps to solve the ...
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204 views

Dangerous substitutions in finding trigonometric integrals.

This is a question NOT on how to actually solve the problem but more about a concern in what one needs to know before making a sort of substitution. Last day I tried to solve this integral: $\int ...
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356 views

Desired Z axis and Yaw to ZXY Euler Angles?

I'm trying to calculate a desired pair of pitch and roll Euler angles (the XY in ZXY format) given a desired z-axis of the rotated frame (expressed in the world frame) and a specified yaw angle ...
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54 views

Rationality of Polynomial Coefficients. Integral Question.

We are entertaining polynomials with roots, all unique, on the curve $\Upsilon_s = \{{( 1-\cos[\theta])^{-s} \exp(i \theta)} \ | \ \theta \in \mathbb{R} \}$, where $s>0.$ $\Upsilon_s$ looks like a ...
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66 views

tangents of infinite sums — reference request

This section of Wikipedia's List of trigonometric identities was, as far as I know, written entirely by me. For a time, it dealt only with finite sums, and said something like this: $$ ...
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169 views

Solve $x \arccos(x)+x/2=\cos(2x)$

$$x \arccos(x)+x/2=\cos(2x)$$ I dont know how to solve this one. It looks relatively easy but it is not, not for me at least. Anybody to help?
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345 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 ...
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449 views

flaw in law of cosines usage

I'm going to through a list of coordinates and computing the angle between every two adjacent lines. In other words, I'm computing an angle for every 3 consecutive points. Every three consecutive ...
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115 views

Inequality with even powers of trigonometric functions

For $m>0$, $0 < n\leqslant m+1$ ($m,n\in \mathbb{Z} $) , and $0 < a < 1$ , prove that $$2^{n}\cdot \left( a^{n}\cos ^{2m}\dfrac {\pi a} {2}+\left( 1-a\right) ^{n}\sin ^{2m}\dfrac {\pi ...
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174 views

References for “closed form” numeric solutions of $\tan x=-a x$

I am looking for references that discuss solutions of the equation $\tan x=-a x$ (for $x,a\in \mathbb{R}$). I know about the graphical approaches, and any number of numerical solution approaches, ...
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191 views

A trigonometric inequality involving sine

Let $0<a<\pi/2,0<b<\pi/2$, $0<\lambda<1, \mu=1-\lambda$. Does anyone see a good proof of the inequality: $$\sin(\lambda a)\sin(\lambda b)+\sin(\lambda a)\sin(\mu ...
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14 views

Length and width of shadow of rectangular plane

A book that I've read shows how to find the area of the shadow cast by a sphere and ellipsoid. The spherical shadow makes sense; its simply the area of a circle (which would be the sphere's shadow) ...
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32 views

Can a sum of trigonometric functions equal a constant for all inputs?

Let $r_1,...,r_n$ and $\phi_1,...\phi_n$ be real numbers. Consider the following sum: $S=\sum\limits_{k=1}^{n}r_k\sin(\phi_k+k\alpha)$ Suppose $S$ is constant for all $\alpha \in R$. Does it ...
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32 views

Mentally approximating an inverse sine?

Is there a method to approximate the inverse of a sine function in ones head? I know one can approximate the inverse of a cosine with the following equation: $\cos^{-1}(x) = ...
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29 views

Find the expected value of the matrix

$\require{cancel}$ I want to see if I have solved this problem appropriately or not. If we have ...
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35 views

Inverse image of $[0,1/2]$ for $f(x)=\sin(x)$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be defined by $f(x)=\sin(x)$. Describe the set: $$f^{-1}\bigg(\Big[0,\frac{1}{2}\Big]\bigg).$$ My answer is $$f^{-1}([0,1])=\bigcup_{n\in\mathbb{Z}} ...
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36 views

Calculating Down Range and Cross Range Coordinates

I have a laser pointer located at: x = 0 in y = -23.53 in z = 50.82 in When pointing the laser in space, I keep the elevation angle fixed but vary the azimuth ...
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20 views

Is there anything significant about the cross-quarter days, in terms of a sinusoid?

As the Earth goes around the sun, the length of the day changes, and certain cultures have celebrations or observances centered around these changes, illustrated in this graph. The red dots, the ...
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22 views

Octahedron and System of trigonometric equations

Could somebody help me to prove the following? $$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \cos (\phi_k)=0$$ ...
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100 views

Riemann sum formulas for $\text{acos}(x)$, $\text{asin}(x)$ and $\text{atan}(x)$

In this post just another $\pi$ formula, I gave a kind of Riemann sum to compute the area of a quarter of circle based on a very simple geometric trick, and same reasoning can be used to compute any ...
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23 views

Function of a point rotating around a moving center

I am currently looking into the following issue: Imagine a point $p$ rotating around a center (0,0). The equations for the coordinates as a function of time are: $x = p_x \cos(t) - p_y \sin(t)$ ...
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44 views

Solving a non-linear system of trigonometric equations

I run into a problem of non-linear system that involve trigonometric components. I need advise to solve then analytically for [x,y]. The equations are \begin{align} & ...
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49 views

Trigonometry Calculate Distance and Angle of object in camera frame

I have an application where I am trying to build a handheld scanner that can draw a 2d profile of a 3d surface (using structured light scanning). The handheld device consists of a line laser and a ...
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16 views

average order of basic trigonometric functions

In connection to a problem in a Fourier Analysis book, I had to calculate the average order of some functions. Then I came across this very elementary yet very interesting observations which follow: ...
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38 views

The zeros of a trigonometric polynomial

Can we describe the set of zeros of a trigonometric polynomial? A trigonometric polynomial is $\sum a_n e^{inx}$. How many zeros can trigonometric polynomials have in an interval with the length ...
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75 views

Proof: Angle between two diagonals of a parallelogram

Can anyone prove the angle $\theta$ (smaller angle) between diagonals of a parallelogram is given by the equation $$\cos(\theta)=\frac{a-c}{a+c}$$ where $2a$ is the base length of parallelogram; ...
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22 views

Tangent identity

I came across this in a paper, and although I suspect its extremely elementary, I can't quite grasp it. Suppose we have vectors $\vec{x}$ and $\vec{x}^{\prime}$. As a given, we have \begin{equation} ...
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12 views

Show if the following inequality holds for an integral of ratio of sin functions

Is it possible to show that, $$\int_{-1/2}^{1/2}\frac{\sin^4(2^n\pi \,f)}{\lvert2\sin(\pi \,f)\rvert^k} \leq \int_{-1/2}^{1/2}\frac{\sin^4(2^{n+1}\pi \,f)}{\lvert2\sin(\pi \,f)\rvert^k}$$ for ...
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30 views

How to find $\sup \Pi_{i=0}^{n} (\sin(i)^2 - \frac{25}{16})$?

Let $\sup,\inf,{\rm dif}$ denote resp supremum , infimum and $\rm dif$ = supremum - infimum. Does any of the 3 below have a closed form ? $\sup \Pi_{i=0}^{n} (\sin(i)^2 - \frac{25}{16})$ $\inf ...
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52 views

How to find $\frac{x}{y}$?

Let $x=\displaystyle\prod_{n=1}^{180}\left(\cos{\left(\dfrac{n\pi}{180}\right)}+2\right)$ and $y=\displaystyle\sum_{n=0}^{89}\binom{180}{2n+1}\left(\dfrac{3}{4}\right)^n$. How to find $\frac{x}{y}$ ? ...
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38 views

Sketch of a possible equivalence with Riemann hypothesis

From Robin's equivalence (see [1]) and the following trigonometrics identitites, I ask to me if it is feasible write vagues equivalences using this strong result and if these equivalences will be ...
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21 views

In a triangle $ABC$, $\tan(A)+\tan(B)+\tan(c)=k$ then how many isosceles triangles exist when $k<0,k>3\sqrt{3}$, and when $0<k<3\sqrt{3}$?

In a traingle ABC $\tan(A)+\tan(B)+\tan(c)=k$ then how many isosceles triangles exist when $k<0,k>3\sqrt{3}$, and when $0<k<3\sqrt{3}$ ? I cant understand how to solve this.Help please..
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19 views

Number of unique solutions to $\sin P_1(x, n_1) = \sin P_2(x,n_2)$

In attempting to answer this question, I was looking at the solutions for $\sin(3x - 4) = \cos(7x)$ when $0 \leq x \leq 2\pi$ (all other solutions should be multiples of these). I found $14$ distinct ...
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40 views

Showing number isn't constructible

How can I show $\cos \left(\dfrac {2\pi}{7} \right)$ isn't constructible? I'm not sure where to begin. I was thinking of using the fact that $\cos \left(\dfrac {2\pi}{7} \right)+i\sin\left(\dfrac ...
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28 views

A 3D problem on the range of angles

$CD$, of length $14$ units, is a line fixed on the ground. $AB$ is a thin rod fixed in the air. A ring, which is attached to $AB$, can slide freely on $AB$. An elastic band (remaining taut all the ...
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37 views

Evaluate Integral of $\int_0^{\frac{\pi 2^n}{2}}\sin^{2+d}(u)\prod_{i=1}^{n}\cos^{2-d}(u/2^i)du$

Integral of $$\int_0^{\frac{\pi 2^n}{2}}\sin^{2+d}(u)\prod_{i=1}^{n}\cos^{2-d}(u/2^i)du$$ I've tried a number of ways to re-write this in a way that makes sense, but no luck thus far. The integrand ...
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62 views

In an acute-angle triangle $ABC$, find the minimum value of $5\tan A+2\tan B+ \tan C$.

Question. In an acute-angle triangle $ABC$, what is the minimum value of $5\tan A+2\tan B+ \tan C$? I have a form : Given $5\tan A+2\tan B+\tan C$ in a $\triangle ABC$ and ...
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33 views

Show that these two identities are equivalent

As an answer to the question Proof that for $a>0$ and $a + 1/a$ element of $\mathbb{Z}$, $a^n + 1/a^n$ is always element of $\mathbb{Z}$ by induction, user236182 gave this answer: ...