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

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71 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 + \sum^...
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64 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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65 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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34 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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27 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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52 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: $\space\...
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53 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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145 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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48 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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51 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, $\frac{x}{\sqrt{1+x^{2}}}+\frac{y}{\sqrt{1+y^{2}}}+\frac{z}{\sqrt{1+z^{2}}}\leq\frac{3\...
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28 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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39 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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38 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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36 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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24 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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44 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 V^E=...
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45 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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48 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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56 views

Parallelogram with vertices $\mathbf{0}$,$\mathbf{Xa}$,$\mathbf{Xb}$,$\mathbf{Xa+Xb}$ ($\mathbf{X}$ matrix, $\mathbf{a}$ and $\mathbf{b}$ vectors)

There is a paralellogram with vertices $\mathbf{0}$, $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{a+b}$, whose area is $34$. What is the area of the parallelogram which has vertices $\mathbf{0}$, $\mathbf{...
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46 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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40 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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42 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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56 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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34 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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61 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}} - \theta_{\vec{a}}))}{|\vec{a}|+|\vec{a}+\vec{c}|}...
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55 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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57 views

Sequences and series of $\tan^n x$

Please help me with this question: Investigate the convergence of the sequence $$\tan x, \quad \tan^2 x, \quad \tan^3 x, \quad \dots, \quad \tan^n x$$ for $x \in (-90^\circ, 90^\circ)$. ...
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47 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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167 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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48 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} &= \sin{y}...
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396 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, 1,...
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382 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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35 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}}; ~~\mathrm{for~0<...
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257 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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56 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) - \cos^2(\theta)...
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33 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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40 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 $B_x$ ...
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62 views

Find: $\sin\left(\frac{2}{\arcsin((x + 4)/5)}\right)$

Find: $$\sin\left(\frac{2}{\arcsin(\frac{x+4}{5})}\right)$$ I know: $$\sin(\arcsin(x)) = x$$ I somehow did something that did get this correct.: $$\sin(2t) = 1 - 2\sin^2(t)$$ So then we see: ...
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64 views

How to simplify sine function

Does anyone have an idea for simplifying this formula? $$f(x)=\prod\limits_{k=2}^{14}\sin(\frac{15x\pi}{k})$$ Or even more general case: $$f(x,y)=\prod\limits_{k=2}^{y-1}\sin(\frac{xy\pi}{k})$$ ...
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231 views

Find Laplace Transform of trigonometric function using unit step function and t-shifting. (5.3-40)

Please check my work. Did I calculate the following Laplace Transform correctly? $$f(t)=sin(t)u(t-\frac{\pi}{2})$$ My solution: Use the following corollary from the second shifting theorem (t-shift)...
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175 views

Calculate vertical lines intersection hexagon at regular interval

I would like to calculate the total size of vertical lines that dissect an hexagon regular, like the one on the image. I would like to know the internal size of the blue lines inside the hexagon, ...
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49 views

From a light house $L$ two ships $P$ and $Q$ are observed in direction South West and $5^{\circ}$ East of South respectively. At same time $Q$…

Question : From a light house $L$ two ships $P$ and $Q$ are observed in direction South West and $5^{\circ}$ East of South respectively. At same time $Q$ is observed from point $P$ in South East ...
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83 views

How to solve sum of sines and cosines system of equations?

I have a set of equations to solve which in the following form: $ \cos(t_1 + t_2 + t_3 + t_4) + \sin(t_1 + t_2 - t_3 + t_4) + \cos(t_1 - t_4 + t_3 - t_5) + \sin(t_1 - t_2 + t_3 - t_5) + \cos(t_1 + ...
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91 views

Conditional inequality

Let x,y,z be positive reals with $xy+yz+zx=1$. Prove the inequality $$\sum_{cyc(x,y,z)}\frac {2x(1-x^2)}{(1+x^2)^2} \le \sum_{cyc(x,y,z)} \frac x{1+x^2}.$$ I substituted $x=tan\frac{\theta}2, y=tan\...
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27 views

Question about Chebyshev Polynomials in Beardon

I happen to be reading through Beardon's book, Iteration of Rational Functions, and I have come across a statement I don't quite believe. He uses it a little later on, so I'm concerned with clearing ...
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97 views

Proving a limit of a trigonometric function: $\lim_{x \to 2/\pi}\lfloor \sin \frac{1}{x} \rfloor=0$

$$\lim_{x \to \frac{2}{\pi}}\lfloor \sin \frac{1}{x} \rfloor=0$$ I need to prove the limit of this using the $\epsilon - \delta $ way but I don't know how to find $\delta$ when I'm given a ...
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134 views

Combinatorial proof for the formula of $\tan n\theta$

Is there any combinatorial proof of the formula for $\tan n\theta$ where $n\in \mathbb{N}$? Then proofs that I know are by Induction and using de Moivre's Formula but recently one of my friend asked ...
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81 views

Did Wolfram Alpha make this term up?

So I'm currently taking a Trigonometry class and I'm using the iOS version of the Wolfram-Alpha app to assist me with some of the more mundane calculations (Pythagorean Theorem, etc) and I've come ...
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147 views

Trigonometry of tetrahedron

I'm trying to develop the algebraic proofs for these two formulas that appear on the webpage below! The image below is of an unfolded non-regular tetrahedron. Triangle B represents the dihedral angle ...
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33 views

Parameterize the equation

Find a way of parameterizing the following curve: $y^2=\sin x $. I have already tried $x(t) = (\sqrt t, \sin^{-1} t) $ but this only gives part of the curve because of the nature of the sqrt function ...