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

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$\sin(2\pi/7) + \sin(4\pi/7) + \sin(8\pi/7) = (root7)/2$

How to do problems such as these 1) $\sin(2\pi/7) + \sin(4\pi/7) + \sin(8\pi/7) = \dfrac{\sqrt{7}}{2}$ 2)$ \sin(\pi/7)\sin(2\pi/7)\sin(4\pi/7) = \dfrac{\sqrt{7}}{8}$
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40 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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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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42 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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40 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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23 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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44 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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42 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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376 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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53 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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24 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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49 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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32 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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31 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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37 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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55 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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66 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 ...
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47 views

Hockey pucks and parameters

There is one hockey puck with a diameter of 3 inches. The puck is spinning around its center at a speed of 3 counterclockwise rotations per second. At the center, the puck is traveling at a speed of ...
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62 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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39 views

What is the best trigonometry book available free?

I am not a rich person but I really want to have a look on the trigonometry book
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25 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 direction of ...
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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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38 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, ...
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21 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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48 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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57 views

How do I find the angle measurement on a triangle that has one curved side?

I have tried taking this from a circle and measuring the angles as if the width and the height are the quarter of a circle however it is not measuring correctly. I have looked at it as if it is the ...
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52 views

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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65 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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77 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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31 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 ...
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49 views

Expectation of product of cosines

I am reading a paper that starts with $$ E[ \cos( a(x-y) ] = E[ \cos(a x) \cos(a y) + \sin(a x) \sin(a y) ] $$ where the expectation is over $a$, then converts it into something of the form $$ = 2 ...
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55 views

Find the length of the longer diagonal on a trapezium with only 2 sides stated.

Im at a loss here, i know i have to divide the trapezium, but im still not sure which calculation is relevant to it then. Thanks in advance.
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31 views

Calculate angle of view from 2D image

I want to calculate the angle of view (or the field of view) from a photograph, without knowing anything about the camera, as to use that information in a 3D environment. I have to use trigonometry ...
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124 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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26 views

Formula for cos((2n+1)x) as polynomial of cos x

I am looking for a formula of cos((2n+1)x) that is polynomial of cos(x). For example, Is it known for any n?
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27 views

Practical determinations of trigonometric identities

I am looking for articles, or any reference, that detail practical determinations of trigonometric identities, with particular emphasis on trigonometric functions raised to the power of 3 or higher. ...
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91 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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112 views

The geometry of a spiral made of adjacent right triangles

In the above figure (not sure if you can see it clearly or not), while using the old standard technique of plotting irrational numbers on number line, I saw this property. If we go on plotting ...
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96 views

Calculating point on sphere surface where sun reflection to a target point occurs

Imagine a mirror sphere at position O with radius R, and a target point at position P, at distance d from the sphere origin. There is an unknown point X on the surface of the sphere, where the light ...
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134 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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Ordinary Differential Equation with a trigonometric function: radius of convergence?

For the equation $$x^2y'' + y' + \tan(x)\,y = 0$$ establish lower bounds for the radius of convergence about the point $$x_0 = 1.$$
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How to find the period of a exponential function? $5\cdot(-1)^k$

Currently in class we are learning Euler's formula and how it relates to imaginary numbers, exponentials, and trig functions. I know the left hand of the equation is $e^{jwt}$ and I could easily use ...
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Nonnegativity on a special domain entails nonnegativity on the whole plane

Let $Q$ be a real bivariate polynomial such that $Q(x,\tan(x))\geq 0$ for any $x\not\in\{\pm\frac{\pi}{2}\}+(2\pi){\mathbb Z}$. Does it necessarily follow that $Q$ is nonnegative on the whole of ...
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61 views

Calculate area of this figure

I have an homework assignment where I have to calculate area of the figure underneath. I used the following formula to calculate the result $\frac {130 \cdot 55 \cdot sin35}{2} = 4188 m$ and then ...
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54 views

Would this thinking about the dot product hold?

Background today I completed the chapter on the dot product of vectors. But in trying to figure out exactly what the dot product is. I came to the conclusion that it can be interpreted as the length ...
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52 views

Using two chords and an angle to find center and radius of a circle

Hello, I am trying to solve the problem below. Is it possible to solve for the Center and Radius of the circle given the information provided, or is there something missing? I know how it's simple ...
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86 views

Proving $\frac\pi{22}\cos\frac\pi{22}+\frac{2\pi}{11}\cos\frac{5\pi }{22}+\frac{2\pi}{ 11}\cos\frac{9\pi}{22}+\frac\pi{22}\cos\frac{5\pi}{11}<\cdots$

$$(\frac{\pi}{22}) \cos (\frac{\pi}{22}) +(\frac{2\pi}{11}) \cos (\frac{5\pi }{22}) + (\frac{2\pi}{ 11}) \cos (\frac{9\pi}{22}) + (\frac{\pi}{22}) \cos(\frac{5\pi}{11}) < (\frac{\pi}{26}) ...
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21 views

Complete Triangle Given 3 Parallel Planes and 2 Points

I have a problem where a point B connects to a point C at a known angle and distance. Both point B and C are on two separate parallel axis, GH and JK respectively. I need to find a third point, A, on ...
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80 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.