2
votes
0answers
29 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
1answer
46 views

Trigonometric regression

What methods are performed for regression with trigonometric functions? E.g. : Sequence: $$-1, 0, 1, -1, 0, 1, \text{.....}$$ Regression: ...
0
votes
1answer
54 views

Strange trigonometric equation

I have to solve the following trigonometric equation: $$2\sin(x)+x\cos(x)=0.$$ This problem comes from an analytical geometry one. I have tried to find the exact solutions, but I didn't succeed. Is ...
0
votes
1answer
97 views

How to calculate a trigonometric interpolation polynomial

I have the following $2 \pi$-period function f: $$ f(x) = \left \{ \begin{array}{l l l} x: & 0 < x < 2 \pi \\ \pi: & x = 0 \end{array} ...
2
votes
3answers
160 views

Law of Cosines for very acute angles, round-off error

We have $$ c^2 = a^2 + b^2 - 2ab\cos(\gamma) $$ If $a \approx b$ and $\gamma$ is very small, then the above formulation has quite a bit of round-off error. Is there a better formulation that would ...
1
vote
1answer
149 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
57 views

shape regular triangulations and Zlamal's condition

I'm trying to show that a triangulation $\tau_h$ is regular if and only if there exists $\theta_0>0$ such that for all $T\in\tau_h$ we have $\theta_T\geq \theta_0>0$, whereas $\theta_T$ is the ...
2
votes
1answer
70 views

Solution to set of three equations

I have the following three equations: $$\cos\theta \left(\cos\psi - k_3\sin\psi\right) = k_1$$ $$\sin\phi\sin\theta\cos\psi - \cos\phi\sin\psi - k_3\left(\cos\phi\cos\psi + ...
1
vote
2answers
212 views

Numerical method to solve a trigonometric (cotangent) function - transient heat transfer problem

I was trying to develop a mathematical model for transient one-dimensional heat conduction of spheres using approximate analytical solution as mentioned in Cengel{refer page number 229 in that ...
7
votes
1answer
309 views

Eigenvalues of a tridiagonal trigonometric matrix

Let $A$ be the diagonal matrix w/alternating in sign diagonal entries: $$ A = \begin{pmatrix} (-1)^{n-1} \tan\left(\frac{\pi}{2n+1}\right) & 0 & 0 & \ldots & 0 \\ 0 & ...
0
votes
2answers
327 views

Trapezoid rule over trigonometric polynomials

The question is regarding trapezoid rule applied on trigonometric polynomials Here is the question Show that the composite trapezoid rule over an equidistant partitioning with interval size $h = ...
0
votes
1answer
44 views

$(a - b \cot \theta) \cos^2 \theta = -\frac{b}{2} \cot \theta$ ,$\theta=$?

This question is a follow up question to this answer. In the equation: $$(a - b \cot \theta) \cos^2 \theta = -\frac{b}{2} \cot \theta.$$ $a$ and $b$ are given. What is the best way to solve for ...
1
vote
4answers
80 views

Solving a set of 3 Nonlinear Equations

In the following 3 equations: $$ k_1\cos^2(\theta)+k_2\sin^2(\theta) = c_1 $$ $$ 2(k_2-k_1)\cos(\theta)\sin(\theta)=c_2 $$ $$ k_1\sin^2(\theta)+k_2\cos^2(\theta) = c_3 $$ $c_1$, $c_2$ and $c_3$ are ...
0
votes
4answers
235 views

loss-of-significance error

Reduce loss of significance error in the following equation by re-arranging terms: $f_1(x) = \frac{1- \cos(x)}{x^2}$ , assuming $x$ is near $0$. Let $f_2(x)$ be the function rewritten to reduce loss ...
1
vote
2answers
158 views

Using Taylor Series for $\sin x$ and $\cos x$ to derive $\cos{(x-a)}$ and $\sin{(x-a)}$

My professor had this problem on our last problem set but got rid of it as it was "more involved" than he thought but I am still curious as to how it would be done (Its good that he ditched it because ...
2
votes
1answer
150 views

Method for finding roots of real trigonmetric polynomial

Given a real valued trigonometric polynomial, $$ f(x) = \sum_{k=0}^{n} a_k \cos(k x + \phi_k) $$ what is the current fastest numerical method to find the roots of the polynomial for a given error? I ...
2
votes
3answers
250 views

Simple test if point is above or below sine curve

Is there any simple formula or algorithm for determining if a point lies above or below the sine curve? For instance, if I have a point $(x, y)$, how can I test whether or not $y > \sin(x)$? ...
5
votes
1answer
636 views

Some approximations for $\arccos(1/(1+x))$

I was trying to calculate the maximum ground distance you can see on mountains, with your elvation given. After some simple geometry, I was able to come up with the following formula: Let $h$ be ...
4
votes
1answer
380 views

Calculation of atan2

I am familiar with the basics of atan2. The doubt I have in the computation of atan2 came across from an image processing sofware. This is a portion of the code segment when x>y. x and y are absolute ...
4
votes
2answers
186 views

Refraction equation, quartic equation

Given two points $P$ and $Q$, a line ($A$, $B$ - orthogonal projection of $P$, $Q$ onto the line) and a coefficient $n$, I want to find out such point $C$ that $\frac{\sin{a}}{\sin{b}}=n$ (in fact, ...
18
votes
8answers
2k views

Rapid approximation of $\tanh(x)$

This is kind of a signal processing/programming/mathematics crossover question. At the moment it seems more math-related to me, but if the moderators feel it belongs elsewhere please feel free to ...
4
votes
0answers
366 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)\\ & {} - ...
5
votes
5answers
2k views

Numerically Efficient Approximation of cos(s)

I have an application where I need to run $\cos(s)$ (and $\operatorname{sinc}(s) = \sin(s)/s$) a large number of times and is measured to be a bottleneck in my application. I don't need every last ...
2
votes
3answers
967 views

Newton's method and trig functions on a computer

I'm trying to use Newton's method to find roots for the function $A \cos(\Theta_2 - \Theta_1) + B \sin(\Theta_1)$. (That is, iterate $x_{i+1} = x_i - f(x_i) / f'(x_i)$). I've got a working ...
0
votes
0answers
179 views

Double integration involving polynomial functions and sinc function

I encountered a problem which I can't seem to simplify/solve. I was wondering if any mathematicians or specialists knows how to approach this problem? $$\int^{0.5}_{-0.5} \int^{0.5}_{-0.5} \; ...
1
vote
1answer
383 views

Fitting a sine function to data

I have a sequence of $n$ points $(x_i,y_i)$, for $i=1,\dots,n$. I would like to find the function, of the form $y=V\sin(x+\phi)$, which best fits the points. Which numerical method could I use? I have ...
7
votes
3answers
602 views

What numerical methods can solve $\sin(x) + \sin(y) = \sin(xy)$

Here is a nice graph representing the solution: wolframalpha. I wish to draw such a graph myself but don't have any idea which methods exist and which of them are more appropriate for equations of ...
1
vote
1answer
877 views

Combining Sine Waves on Chart

I would like to combine multiple sine waves with differing amplitudes, frequencies and phases into a single curve that I can display as a graph. What formula will I need to create the points for the ...
2
votes
2answers
287 views

Solving short trigo equation with sine - need some help!

From the relation $M=E-\epsilon\cdot\sin(E)$, I need to find the value of E, knowing the two other parameters. How should I go about this? This is part of a computation which will be done quite a ...