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

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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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41 views

Solve $x/(4x^2+1) = \tan(6x)$ for $x$

$$ \frac{x}{4x^2+1} = \tan(6x) $$ Can this equation be solved for $x$ algebraically and can I get exact answer for this question? Or do I have to approximate it?
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234 views

How to easily prove Euler's theorem, $OI^2=R(R-2r)$?

If $R$ is the circumradius and $r$ is the inradius of some triangle $ABC$, with its circumcenter being $O$ and incenter being $I$, then how to prove: $$OI^2=R(R-2r)$$ I have seen many mentions of ...
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120 views

Launch angle required to hit coordinate (x,y) with air resistance

Finding Angle of Elevation to hit X, Y and Wikipedia Angle required to hit coordinate work, but don't calculate air resistance. Is there a way to find the launch angle of a projectile required to hit ...
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98 views

Application of integrating $\cos^4 x$?

A student asked a colleague the other day for a practical application that involved needing to integrate the fourth power of cosine, but no one here could think of one off-hand other than some volume ...
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69 views

Can one generate all possible binary strings by sampling a trig function at regular intervals?

I'm using a trigonometric function to generate binary strings by sampling the function at regular intervals and mapping each sample value to a binary bit. As a simple example: if the function is ...
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143 views

Trigonometric curiosity

How prove this $$-\tan\frac{10\pi}{41}+4\left(\sin\frac{2\pi}{41}+\sin\frac{4\pi}{41}+\sin\frac{12\pi}{41}+\sin\frac{20\pi}{41}-\sin\frac{26\pi}{41}-\sin \frac{30\pi}{41}\right)= ...
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54 views

Polygons inscribed in circles, with integer sides and integer radius

Is there a simple characterization for an integer partition $(s_1,\dots,s_k)$, such that a polygon with these sides is inscribed in a circle with integer radius? This is what I got so far: All ...
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202 views

How to find the approximate basic frequency or GCD of a list of numbers?

I could't actually summarize the question in the title, so I'll explain my situation. I want to tell the integer numbers which act as the best approximate basic frequencies of a list of real numbers: ...
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107 views

Evaluating $\int_0^x \lvert \cos t \rvert dt$

in my mathbook there is given a solution to $$\int_0^x \lvert \cos t \rvert \, dt $$ but without any hints or tips. $$\int_0^x \lvert \cos t \rvert \, dt = \sin\left(x - \pi \left\lfloor \frac x ...
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72 views

simultaneous trigonometric equations

Consider the pair of simultaneous equations $p_5\cos(2\omega\tau)+p_4\omega\sin(2\omega\tau) = p_1\omega^2-p_3-p_6\cos(3\omega\tau) $, $p_4\omega\cos(2\omega\tau)-p_5\sin(2\omega\tau) = ...
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94 views

Period of trigonometric function

What is the period of $$\frac{7\sin x + 5\cos x}{7\sin{2x} + 11\cos x}$$ What should I do here? I don't even know where to start from. Please help me by giving me a hint!! Thanks.
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54 views

From $\tan(1/A) = \tan(1/B) + \tan(1/C)$ to $A + B + C = ABC$

In this recent question, the equation $$\tan\left(\frac{1}{A}\right) = \tan\left(\frac{1}{B}\right) + \tan\left(\frac{1}{C}\right)$$ is said to imply $$A + B + C = ABC$$ without any stated ...
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90 views

Eliminate variable in trigonometry equations

Say you have the equations: \begin{align} -S_1\sin\left(2\psi+\theta\right)+S_2\cos\left(\psi\right)&=S_3\\ S_1\cos\left(2\psi+\theta\right)+S_2\sin\left(\psi\right)&=S_4 \end{align} or ...
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76 views

Solving systems of equations with trigonometric terms

I am trying to solve (or rather find the least squares solution for) a system of equations with trigonometric terms in them. The system consists of pairs of equations of the form $a_1 \cos\theta - ...
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227 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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69 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 ...
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1k 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 ...
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44 views

Finding the sum of a trigonometric series, fourier series

I need to compute that for $x \in [0, 2\pi]$ $$\sum_{n=1}^\infty\frac{\sin(nx)}{n^3} = \frac{1}{12}x(x-\pi)(x-2\pi)$$ by using the uniform convergence $$\sum_{n=1}^\infty\frac{\sin(nx)}{n} = ...
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163 views

“Straightforward” application of trigonometry to the slingshot effect/gravity assist

I have been trying to understand the formula $$v_f^{2}=v_i^{2}+2V(V(1-cosβ)+v_i(cos(α-β)-cosα))$$ as it relates to Fig. 2 on page 5 of this exposition: ...
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46 views

Is there a way besides integration by parts to solve this integral?

$$\int_{0}^{2\pi} -10\cos^9(t)\sin^4(t)t^4\,dt$$ Maybe a formula for this form or something?
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29 views

Solving inequalities of trigoniometric function of multiple variables

How does one solve an inequality such as the following: $\sin^{2}(x+y) > \sin^{2}(x) + \sin^{2}(y)$ I can do this: $(\sin(x)\cos(y) + \sin(y)\cos(x))^{2} > \sin^{2}(x) + \sin^{2}(y)$ ...
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35 views

integrate the square of angular distance from the node of a spherical triangle

Guessab, Noouisser, and Schmeisser "A Definiteness Theory for Cubature Formulae of Order Two", Constructive Approximation (2006)24:263-288 Define a quantity $R[||\cdot||^2]$ which is $$\sum_{i=1}^N ...
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293 views

($\cos^4x$)($\sin^2x$) in terms of first power of cosine

I believe that I have his correct but if someone could check it and see that'd be great. Here's a pic!
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52 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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96 views

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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144 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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205 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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35 views

Calculate the following expression

Calculate the value of $\sin a+\cos a$, knowing that $\sin a\cos a=0.48$ and $a$ belongs to $\left[\pi; \frac{3\pi}{2}\right]$? What I did is: $$\sin a \cos a=0.48\ \ |\times 2$$ $$\sin 2a=0.96$$ ...
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44 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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73 views

An other tricky one. Tigonometric integral.

How should I attack this ? $$ \int_0^{\pi} \cos(ax)^m \cos(x)^n dx$$
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119 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

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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96 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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106 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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89 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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469 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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88 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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59 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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112 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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281 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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106 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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115 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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686 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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205 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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390 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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145 views

Proof of extrema of $\sin(x)$

I need to find the values of $a$ at which $$\lim_{h \to 0} \frac{\sin(a+h)-\sin(a)}{h} = 0.$$ I know that this means that we are looking for the values of $a$ at which $\dfrac{d}{dx}\sin x=0$, or ...
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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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179 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?