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

learn more… | top users | synonyms (2)

2
votes
0answers
199 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: ...
2
votes
0answers
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 ...
2
votes
0answers
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) = ...
2
votes
0answers
93 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.
2
votes
0answers
53 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 ...
2
votes
0answers
89 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 ...
2
votes
0answers
74 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 - ...
2
votes
0answers
223 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 ...
2
votes
0answers
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 ...
2
votes
0answers
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 ...
2
votes
0answers
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} = ...
2
votes
0answers
160 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: ...
2
votes
0answers
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?
2
votes
0answers
28 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: $(\sin(x)\cos(y) + \sin(y)\cos(x))^{2} > \sin^{2}(x) + \sin^{2}(y)$ ...
2
votes
0answers
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 ...
2
votes
0answers
286 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!
2
votes
0answers
51 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: ...
2
votes
0answers
92 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 ...
2
votes
0answers
142 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 ...
2
votes
0answers
204 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 ...
2
votes
0answers
34 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: $$\sin a \cos a=0.48\ \ |\times 2$$ $$\sin 2a=0.96$$ ...
2
votes
0answers
41 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: ...
2
votes
0answers
73 views

An other tricky one. Tigonometric integral.

How should I attack this ? $$ \int_0^{\pi} \cos(ax)^m \cos(x)^n dx$$
2
votes
0answers
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} ...
2
votes
0answers
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 ...
2
votes
0answers
95 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 ...
2
votes
0answers
105 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 ...
2
votes
0answers
87 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 ...
2
votes
0answers
462 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 ...
2
votes
0answers
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 ...
2
votes
0answers
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 + ...
2
votes
0answers
109 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} = ...
2
votes
0answers
278 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 ...
2
votes
0answers
105 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 ...
2
votes
0answers
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 ...
2
votes
0answers
675 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 ...
2
votes
0answers
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 ...
2
votes
0answers
383 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 ...
2
votes
0answers
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 ...
2
votes
0answers
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 ...
2
votes
0answers
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: $$ ...
2
votes
0answers
177 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?
2
votes
0answers
359 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 ...
2
votes
0answers
504 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 ...
2
votes
0answers
117 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 ...
2
votes
0answers
182 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, ...
2
votes
0answers
193 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 ...
1
vote
0answers
37 views

Integral involved trigonometric funtions

I try to find closed form solution or approximate function to the following integral: \begin{equation} \int \!{\frac {{x}^{2} \cos^{2} \left( x \right) +x\sin \left( x \right) \cos \left( x \right) ...
1
vote
0answers
13 views

How to determine if the $\theta_i$ located in the interval?

I have a interval $I:=[\pi -\theta_T,\theta_T+2\pi]$, where, $\theta_T \in [-\frac{\pi}{2},\frac{\pi}{2}]$ In addition, there are some points $\theta_i \in [-\pi,\pi]$ where, $i=1,\cdots , 4$ So ...
1
vote
0answers
17 views

How can I find the distance between two points within a triangle if I have the distance between each point and each vertex of the triangle?

Title says it all. It would be useful to extend the question to finding the distance if any of the points is outside of the triangle, but I'm trying to figure out the basic problem first.