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

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307 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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385 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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114 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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168 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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181 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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18 views

How are closed form solutions for eigenvalues constructed from sines and cosines?

For example a size N 3-point finite difference scheme has eigenvalues $\lambda_j = 2+cos(j\pi/(N+1))$. How is this determined? I know the Gershgorin circle theorem, but this is not what I am looking ...
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17 views

Derivation/equation for solid angle factor correction

Derivation/equation for solid angle factor correction Summary: I want to determine a correction for the Solid Angle Factor (SAF) due to partially overlapping 'outer' spheres (of different sizes), as ...
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40 views

Standard properties of trigonemetric functions

You have $\sin(\frac{\pi}{6})= \frac{1}{2}$ and $\sin(\frac{5\pi}{6})= \frac{1}{2}$ and the interval [$\frac{\pi}{6},\frac{5\pi}{6}$] has length $\geq 1$ This is used as I'm sure most will be familiar ...
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24 views

Cosine formula to show if an angle is obtuse or acute

Keeping in mind the cosine formula: $a^2 = b^2 + c^2 -2bc\cos A$, or rearranging $\displaystyle{\cos A = \frac{b^2+c^2-a^2}{2bc}}$, I need to show when $A$ is acute and when it is obtuse. Consider ...
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34 views

Solve:$\int_{0}^{t}{{\left(\cos({…})+\sin({…})\right)} \frac{\lambda^2 e^{(…)}}{\sqrt{\pi (t-r)}} \text{Erfc}{\left(… \right)} }~\mathrm{d}r$

I have another nasty integral to solve as follow: $$ I(t)=\int_{0}^{t}{{\left(\cos({\frac{\gamma}{4(t-r)}})+\sin({\frac{\gamma}{4(t-r)}})\right)} \frac{{\lambda^2} e^{2 \lambda^2 r+ \lambda ...
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32 views

Integral of $|\cos(ax))|\times e^{-x^2/b}$

I can compute the following integral very easily ($a$ and $b$ are real and positive): $$\int_{-\infty}^{\infty} \cos(ax)\times \frac{1}{\sqrt{\pi b}}\cdot e^{-\frac{x^2}{b}}\,dx = ...
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56 views

How do you find the Fourier series of $\max(0, \sqrt{1 - \cos{\theta}})$?

I was trying to express the following periodic function: $$ f(x) = \max \left( 0, \sqrt{1 - \cos{x}} - \frac{\sqrt{2}}{2} \right)$$ as a summation of cosines and sine waves $f(x) \approx a_0 + ...
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53 views

Integration of certain real functions using Euler's Formula.

I've heard about using Euler's formula $$e^{ix}=\cos(x)+i\sin(x)$$ to transform rational functions of sine and cosine into computable indefinite integrals. However, upon attempting to apply this ...
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61 views

How $|x|<a\implies a>0$

The title is not exactly what I'm asking, so sorry for that. I was doing a problem in my mathematics text book. It is given that $|x|<a$, I thought if $a=2$ then we can put $x=1$ but what if ...
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18 views

Computer game Dev, translate rotated camera coordinates to game world coordinates.

bit hard to title this question. Hopefully this is the right exchange to ask it on. Anyways, to the question! So I'm developing a game for ios and am having a few problems getting the maths correct ...
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26 views

How to use 2D Translation and Rotational error to get offset value for new point?

Here I am trying to detect FIDUCIAL points on PCB in real time using camera. After googling for Two days and reading many post and blog. I found that I have to do something called translational error ...
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22 views

$ABCD$ has area $9$. $M$ is in the middle of $AB$ and the edge $BF$ of length $2$ forms an angle of $60º$, Calculate $[CM,CB,BF]$.

$ABCD$ has area $9$. $M$ is in the middle of $AB$ and the edge $BF$ of length $2$ forms an angle of $60º$. Calculate $[CM,CB,BF]$, knowing that $\mathbb{V}^3$ is oriented by a positive basis. ...
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43 views

Simplifying cyclometric function

How does one simplify this function? $$ f(x) = \arccos(\frac{\pi}{2} - \sin(x)) $$ A plot in GeoGebra showed a graph that looked like semicircle, so can one expect something in this form: ...
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39 views

Complex numbers and simple argument question

Yesterday, i encountered a question: $z=a+bi$ $Arg(z-\overline z + 4) = {4\pi \over 3}$ $b=?$ I solved the question using basic method: $$\overline z = a-bi$$ $$ w = z - \overline z + ...
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37 views

Trigonometric identity reduction

I want to be able to reduce some trigonometric expressions that have powers of sine and cosine. For example, for arbitrary real numbers $a$, $b$, and $c$, we can reduce the expression $$ a\cos^2\theta ...
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64 views

Law of Cosines for SSA triangles

In most geometry courses, we learn that there's no such thing as "SSA Congruence". That is, if we have triangles $ABC$ and $DEF$ such that $AB = DE$, $BC = EF$, and $\angle A = \angle D$, then we ...
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44 views

L'Hôpital's rule exercise concerning trig function

I'd like to verify that my work on the following L'Hôpital's rule question is correct: $$\lim_{x \to 0}\,\,{\cot{x}\,(x^2+3x)} $$ As the limit evaluates to $\frac{0}{0}$, we take the derivative of ...
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46 views

proof of inequality perhaps using trigonometric identity

I need help on the following problem. Let $x,y,z$ be the positive real numbers and satisfy $x+y+z=xyz$ then, ...
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75 views

3D surface intersections

I tried to look at 3D Hypersurface intersections of 4D this way based on four Mathematica (circular) trigonometric parametrization combination selections. No hyperbolic functions are directly ...
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24 views

Finding the value of a trigonometric equation

Find the numerical value of $$\tan(3\pi/11)+4\sin(2\pi/11)$$ without actually calculating the values. How to start? Please help.
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30 views

Realistic Bounce (Using Trig?)

background: I am making a graphics program where the major purpose of it is to have a ball (traveling on an arbitrary slope) to bounce realistically off of a line (which is also at a arbitrary slope). ...
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30 views

Converting solutions to separation constant to Cosh and Sinh

The Laplace's equation inside a rectangle is: $$u_{\text{xx}}+u_{\text{yy}}\text{=0}$$ The IC's are: $${u(0,y)=g(y)}$$ $${u(L,y)=0}$$ $${u(x,0)=0}$$ $${u(x,H)=0}$$ Via method of separation we have ...
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24 views

Proving that $\sin(nx) = \sum_{j=0}^{[n/2]} (-1)^j {n \choose 2j}(\cos(x))^{n-2j}(\sin(x))^{2j}$ with induction

We have to prove: $$ \sin(nx) = \sum_{j=0}^{[n/2]} (-1)^j {n \choose 2j}(\cos(x))^{n-2j}(\sin(x))^{2j}$ with induction $$ where $[n/2]$ stand for the floor function of $n/2$. I know this formula can ...
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20 views

Is it possible to derive circumference from these two points?

I have two points along one axis, call it y. I don't have the x axis coordinate because the points were taken as 1-D measurements. The angle between the points is known. Is it possible to derive a ...
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36 views

Using axis coordination to represent rotation matrix instead of angles

Euler angles give us clear matrix for conversion of a vector from car reference $Fr^C$ to earth reference $Fr^E$. If $\vec V$ is a vector in different frames it is represented differently: $$\vec ...
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21 views

Calculating pairwise distance of two N-dimensional vectors given their length and angle

I am not a mathematician, so apologies in advance for any nomenclature blasphemy. Given the magnitudes of two vectors $b$ and $c$ and the angle between them $A$, I can calculate their distance in 2-D ...
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44 views

Is this simplification 'allowed'?

I've just been doing a problem that involved this equation: $$ \frac{1}{\sin\left(\frac{\theta}{2}\right)}\left( \sin\left(b\theta-\frac{\theta}{2}\right)-\sin\left(a\theta-\frac{\theta}{2}\right) ...
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29 views

Parallelogram with vertices 0, Xa, Xb, Xa+Xb (X is matrix, a and b are vectors)

There is a paralellogram with vertices 0, a, b, and a+b, whose area is $34$. What is the area of the parallelogram which has vertices 0, Xa, Xb, and Xa+ Xb, where X = \begin{pmatrix} 3 & -5 \\ -1 ...
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44 views

Prove that these result are the same

I did this trigonometric integral in two different ways, and the results that I got were with two different trigonometric functions, $\sec x$ and $\tan x$. The integral is: $\mathbf{\int tan^{5}x \, ...
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34 views

Confused about integration over zeroes.

Does for example $\int_{-\pi}^{\pi} \sin(x) \, dx$ cancel out to zero (following WolframAlpha/normal integration technique), or do we have to take the absolute value of all the areas between bounds ...
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31 views

Can you always cover a circle in a finite number of steps with this “radar” algorithm?

Suppose you have a disc $C$ of radius $V$ with center $c$ and you randomly place a point $p$ in it. $p$ Behaves as follows: at every time-step, $p$ calculates its angle to $c$, and moves a distance of ...
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46 views

Combinations of Chebyshev polynomials and sin functions

By chance, I see this formula $\int_0^1 T_{2n+1}(x)\sin(ax) { dx \over \sqrt{1-x^2}}=(-1)^n\frac{\pi}{2}J_{2n+1}(a)$ but what is the closed form if we have $\int_0^1 T_{2n}(x)\sin(ax) { dx \over ...
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23 views

Workout line segment inside expanding circle

I have what is probably a fairly basic math problem for a game I'm creating. On each frame I need to work out how much a sub segment of a line passing though a circle will expand when the circle ...
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47 views

Hints to find analytical solution to integral

I have to evaluate the expression $$f(|\vec{c}|) = \int_0^\infty \int_0^{2\pi} (z(\vec{a})+z(\vec{a}+\vec{c})) \frac{(1-\cos(\theta_{\vec{a}+\vec{c}} - ...
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46 views

Can trigonometric equations be graphed?

I was solving various trigonometric equations. I was confused that how are they solved easily by using methods that are useful to solve algebraic equations. Do the trigonometric functions in ...
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29 views

Calculate the overall circle enclosing multiple smaller circles

I have multiple smaller circles of a fixed radius that I am using to define a larger enclosing circle. So I'll need to find the x and y and radius of this new circle. I am looking for efficient over ...
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78 views

How to find XYZ Coordinates of the Major and Minor Axis end-points in an orbit?

To give some context to this problem, I'm attempting to convert an orbit into a Cubic Bezier Spline, by first plotting four points around the Orbits Ellipse and then computing the Control points of ...
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43 views

Prove $\frac{d}{dx}[\csc^{[-1]}{x} = \frac{-1}{|x|\sqrt{x^2-1}}$

Show that the $\frac{d}{dx}[\csc^{[-1]}{x} = \frac{-1}{|x|\sqrt{x^2-1}}$. $$ \begin{align*} y &= \csc^{[-1]}{x} \\ \csc{y} &= x \\ \frac{1}{\sin{y}} &= x \\ \frac{1}{x} &= ...
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389 views

Count points on x-axis

Given S and C . There are S sine functions and C cosine functions as following: $F(i,x)$ = $sin(2^i x)$, $0 ≤ x ≤ 2π$, for $i = 0, 1, ..., S−1$ $G(j,x)$ = $cos(2^j x)$, $0 ≤ x ≤ 2π$, for $j = 0, ...
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106 views

Converting Pixel displacement to radians or mm

How do i convert a pixel displacement to a displacement in radians, or mm.. I need the formula to convert to a program, for which i know the displacement in pixels, but need it in radians.
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27 views

Sine and Bessel integral extension to imaginary argument

I found this integral in Gradshteyn-Ryzhik's book, $$ \int_a^\infty\ J_0\left(b\sqrt{x^2-a^2}\right)\ \sin(cx) \mathrm{d}x = \frac{\cos\left(a\sqrt{c^2-b^2}\right)}{\sqrt{c^2-b^2}}; ...
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89 views

Derivative of angle between two vectors singularity!

I have been battling a problem of needing to know the derivative of the angle between two vectors, the vectors possibly being parallel at some points in time. I started off with: $$\bf A \dot \bf B = ...
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33 views

Solving trig functions with graphing calculator

I know that $ \sin^2(\theta) + \cos^2(\theta) = 1$, but I am not sure how to verify this with a graphing calculator. I am using a TI-Inspire CAS. I also want to find $ \sin^2(\theta) - ...
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32 views

Geometry problem: relation between side of an equilateral triangle and that of a regular heptagon

I need to prove that half the side of an equilateral triangle inscribed circle differs from side of a regular inscribed heptagon by less than $\dfrac{1}{500}$ of the radius. I am stuck and couldn't ...
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38 views

Solving $q = \sin\left(\frac{a}{2}\right)*\cos(B_x)$ for $B_x$

I'm a bit rusty and am having trouble using Trig Identities to solve for $B_x$. Can someone show me how to do this? $$q = \sin\left(\frac{a}{2}\right)*\cos(B_x)$$ I want to solve for $B_x$ ...