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

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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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44 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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66 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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21 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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27 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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29 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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23 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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35 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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16 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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43 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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43 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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23 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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42 views

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

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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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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21 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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43 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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45 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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26 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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57 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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381 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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82 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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26 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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65 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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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 $B_x$ ...
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56 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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84 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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59 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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75 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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52 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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28 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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44 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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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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51 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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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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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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71 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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87 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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53 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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92 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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38 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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136 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?