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

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111 views

What would be a close equation representation of this repeating line pattern?

A quick observation might conclude that this is just a sin function, but the thing I'm looking to find the answer to is the straightness between each maximum, and brief dip before and after the ...
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135 views

Eliminating $\theta$ from the system

That's a modified exercise taken from a admission test to a university. Let $x$, $y$ and $\theta$ real numbers such that $$\left \{ \begin{array}{l} x\sin \theta + y \cos \theta = 2a \sin 2\theta\\ x ...
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313 views

find shortest length of a isosceles trapezoid

Im not sure if this question is easy or not, the concept of what I'm asking seems simple but I cant figure it out. Given that A0 = 100, and h = 10, how would I calculate Az. I have begun by ...
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67 views

Help with manipulating a change of variable in Integration

Knowing: $$\phi (x)=\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { dt }{ \sqrt { 1-{ x }^{ 2 } \sin ^{ 2 }(t) } } } $$ I am trying to demonstrate that: $\phi (x)=\frac { 1 }{ 1+x } \phi \left( ...
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55 views

Simple equation misunderstanding

Im trying to use an equation on this page http://en.wikipedia.org/wiki/Bicycle_and_motorcycle_dynamics The angle of lean, $\theta$, can easily be calculated using the laws of circular motion $$ ...
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64 views

Why the cosin of an acute angle in a right-angled triangle

That is (the cosin I mean), the abscissa of a generic point P along the circumference or the lenght of the perpendicular projection of P onto the ordinate, is given by the ratio of the adjacent ...
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82 views

referenence request: yet another tangent half-angle formula

It is widely known that $$ \frac{\sin\alpha+\sin\beta}{\cos\alpha+\cos\beta} = \tan\frac{\alpha+\beta}{2}. $$ I'm wondering if the following is "known" in the sense of being in published sources? ...
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97 views

Bounding derivative of a function

Consider $a(t)\in\mathbf{L}^{2}(\mathbb{R})$ and $a(t)>0$, is a low pass smooth function with $\hat{a}(f)=0, |f|>f_{max}$. Can we have a upper bound on the following, ...
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119 views

Solve $x \arccos(x)+x/2=\cos(2x)$

$$x \arccos(x)+x/2=\cos(2x)$$ I dont know how to solve this one. It looks relatively easy but it is not, not for me at least. Anybody to help?
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169 views

Apply a Yaw to Pitch and Roll

I have a unit vector of two angles. A roll, rotation around Y-axis; and a pitch, rotation around X-axis. If I apply a yaw, rotation around the Z-axis, how do I calculate what the new angles are of the ...
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75 views

Definition of Inverse Cotangent

I would like to derive the following expression for inverse cotangent: $\cot^{-1} (z) = \frac{i}{2}[\ln(\frac{z-i}{z}) - \ln(\frac{z+i}{z})]$ But I don't want to take it as "definition" as this page ...
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140 views

Solving or approximating an equation with radicals and arctan function

I have solved a differential equation recently, which left me with this whopper of inverse function to figure out. I know what $c$ is, I just haven't calculated its exact value based on the initial ...
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199 views

Derive trigonometric functions from these equations.

(A) $\sin(-x)=-\sin x$ (B) $\cos(-x)=\cos x$ (C) $\cos(x+y)=\cos x\cos y-\sin x\sin y$ (D) $\sin(x+y)=\sin x\cos y+\cos x\sin y$ Use these equalities to derive the following important ...
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1k views

Direction ratio of a vector in 3d?

Suppose I have a vector whose starting end is the origin & the other end can be in any of the 8 quadrants (in 3d). I can easily get direction ratios for the vector & also know in which ...
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3k views

A complicated trigonometric equation

I have the following trigonometric equation $$f(\theta)=100(A_2 B_3 - A_3 B_2)^2 - (c_1B_3 - c_2 B_2)^2 - (c_2A_2 - c_1 A_3)^2=0,$$ where: $ A_2 = 3\cos(\theta)-5$ $B_2 = 3\sin(\theta)$ $A_3 = ...
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210 views

Calculating the Epsilon Neighborhood of line segments in 3d

I am working on a trajectory clustering algorithm (in C++) and one of the steps required in this algorithm is to take a set of 3d line segments (D), and for each line segment (L) in D, to calculate an ...
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486 views

How do I calculate the cartesian coordinates of stars

Given the Right ascension in h m s, Declination in deg ' " and the Trigonometric parallax How can I get the cartesian (x,y,z) coordinates of a star? I'm guessing I need 3 separate formulas to get each ...
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15 views

Collision between 2d circle and flat surface

First of all I want to preface this post by saying that I am absolutely terrible at maths, my level of geometry equals being able to discern a circle from a rectangle but that's about it, as for ...
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8 views

How to find optimal perpendicular axis of rotation vector?

I am drawing lines on the screen. Each line has a point (x,y,z) and a direction (u,v,w). I want to draw arrow heads on these ...
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23 views

Find all the angles between $0^\circ$ and $360^\circ$ which satisfy the equation.

Find all the angles between $0^\circ$ and $360^\circ$ which satisfy the equation $$\tan 3x-3 \sin 30^\circ=0$$ I tried searching for examples but didn't get any. Please teach me how to solve such ...
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24 views

Real world tangent functions

I am a high school math teacher and one of my students asked me for examples of real world tangent functions. Not using tangent to find a side length but a relationship that can be represented by a ...
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32 views

Finding the inverse of trig functions

I'm supposed to find the inverse of $$f(x) = \cos(x)+x$$ I usually just substitute $x$ for $y$ and then re-arrange. What do I do in this scenario?
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32 views

Given a non-negative integer $m$ and a positive integer $n$, calculate $\lfloor \frac{m}{n} \rfloor$

Here is the problem: I have a non-negative integer $m$ and a positive integer $n$ I would like to calculate $\lfloor \frac{m}{n} \rfloor$, $\lceil \frac{m}{n} \rceil$ and $m \bmod n$ But I want to ...
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29 views

Phase vocoder equation

I know I can adjust the frequency of a waveform using a modified version of the sine wave equation amplitude*cos(2*pi*frequency*time+phase) this will allow me to adjust the frequency of a signal. ...
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14 views

Solution to Tanθ = -3/4 in converting to cylindrical coordinates

I am attempting to convert (8, -6, 7) from rectangular coordinates into cylindrical coordinates. We have r = 10, but then I end up with tanθ = -3/4 and I am not sure how to get an exact answer for ...
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39 views

An isosceles triangle with a measure of the sides abc of $5,5$ and $4$.

Find the angles of the triangle in an isosceles triangle of length 5 as the hypotenuse and $\sqrt{21}$ as the height of the triangle as well as the angle bisector, and measure the angles and find ...
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34 views

Polygones inscribed with in a circle

Let's say that there is a circle in two dimension and the diameter of the circle is 1.First start with an equilateral triangle inscribed with in the circle and the measure of the angles are equal to ...
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15 views

Trig Star problem

I need to find an angle where I have a radius of 50. The Radius starts at point A which Forms a 90 degree angle CAF the distance between C & F is 70.71. The angles For ACF and AFC are both 45 ...
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13 views

What is the domain of complex tangent function?

What is the domain of $\tan z =\frac{\sin z}{\cos z}$ ? Is the domain $D=\{z\in\mathbb{C} : \cos z \neq 0\}$? Or considering the Riemann sphere, is $\tan z$ defined on $D$ as $\infty$?
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44 views

How and why is a trig equation different from a regular equation?

$\frac{\sin(x)}{\cot(x)}+\cos(x)=\sec(x)$ is a example for the question. When proving this identity work for all values for $x$, you will end up with either $\frac{1}{\cos(x)}=\frac{1}{\cos(x)}$ or ...
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20 views

area of surface Function for Champagne flute

Can any body tell me how to solve number 5 on this page i have attached, finding a function for champagne flute while function for wine glass is given.
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35 views

Stuck with trig substitution

I am stuck with problem at my homework assignment. $$\int \sqrt{1+4x^2}dx$$ I try to apply trigonometric substitution $$x = \frac 1 2\tan{2u}$$ $$dx = \frac 1 {\cos^2{2u}}du$$ But after ...
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19 views

linear/nonlinear system definition

genmerally we know that linear system is defined by following two rule $1.T(x+y)=T(x)+T(y);$ $2.T(c*x)=c*T(x) $ or operation on sum of two income is equal to sum of operation on each input and ...
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14 views

Find semimajor axes of ellipse from two points and normals

I have two points on an ellipse and normals perpendicular to the ellipse for each. I know where the vectors intersect and the angle between them. How can I compute the length of the ellipse axes? ...
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10 views

Relation of Product Rule to sine sum identity

I was taking a look at the sine sum identity and noticed a resemblance to the Product Rule for derivatives. Applying this led to the following simplification: $$\begin{align}\sin(x + y) & = ...
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26 views

I stumbled when i saw this (Travelling Salesman related)

I have here 5 locations like below I then have a tour of 1,2,3,4 (point 5 isn't inserted into tour yet) like below I then find the shortest addition distance to include point 5 into tour, the ...
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30 views

Find the solutions of the equation…

Can you help me to solve this equation? This is the equation. I understand that the point in which $\sin$ is 1 is $\frac{\pi}{2}$ but I know that there are a lot of this point like $\frac{3\pi}{2}$ ...
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23 views

Determine the 3d position of this vector

I have a rather simple mathematic problem to solve. However I am not that mathematically inclined so I'd appreciate if someone could provide advice on how I solve this mathematic problem: For the ...
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27 views

Trigonometric Power Formulas (or something more modest)

How does one begin to show (natural $n$): $$\cos^{2n}(x) =\frac{1}{2^{2n}} \binom{2n}{n}+ \frac{1}{2^{2n-1}} \sum_{k=0}^{n-1} \binom{2n}{k} \cos[2(n-k)x]$$ $$\cos^{2n+1}(x) =\frac{1}{4^{n}} ...
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13 views

Need help with a trigonometric conversion exercise.

$$\ \theta=\frac{G^º}{12}+\frac{2^g}{D}+\frac{rd}{36}rad $$ Calculate minimum positive value for angle θ measure Edit: I know how to convert from radians to grads but I don't understand the problem
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23 views

Prove that the given triangle is isoceles

I'm trying to solve a problem here. It says: "Prove that a triangle is isoceles if $\large b=2a\sin\left(\frac{B}{2}\right)$." $B-beta$ I've tried to prove it but I can't Can anyone help me?
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18 views

limitations of non linear multivariant equation solvers

I have a system of non-linear multivariate equations. I am only interested in the roots of the system in an interval of each variable. For example, $$ \begin{align} \frac{1}{10} \sin \left( \frac{x ...
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25 views

Triangle rotating freely around origo, need to calculate corners.

Lets say I have a triangle with corners $(-1, -2)$, $(0, 2)$ and $(1, -2)$. I specify a line that is exactly one side of the triangle, for example $(-1, -2)$, $(0, 2)$. Now, I rotate the triangle ...
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59 views

Finding a general solution for trigonometric equations

I'm trying to finish this practice paper and need help finding the general solution. This is the task: "Maths End amusement park has two Ferris wheels: the Kiddy-wheel, a small wheel that reaches a ...
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15 views

Shifting, reflecting, translating the graph of $y = \sec x$.

So i have a question like this... Write the equation for the following curve in its final position The graph of $y = \sec(x)$ is shifted $\pi/4$ units to the left, reflected across the x-axis, then ...
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13 views

Inverse ease-in-out parametric function

I'm trying to create an inverse ease-in-out function that given values from 0 to 1, produces values from 0 to 1. Opposite of a typical ease-in-out function, though, I want it to start accelerated, ...
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30 views

Simplifying trig through geometry

I have the expression $$\arctan{\left(\frac{\sqrt{b}}{\sqrt{a-b}}\right)}$$ and am wondering whether there is any way to simplify this further? Geometrically we have a triangle but that's as far as I ...
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22 views

Bin Packing Algorithm and Minimal Area Axis-Aligned Bounding Boxes

I am a computer hobbyist and, just for the heck of it, I have decide to work on a bin packing algorithm. I would like for the program to eventually handle complex 2-D objects with bezier curves and ...
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53 views

Find points of intersection with cone on a plane at a given angles

The provided variables are the cone angle(cA) of a cone that starts at the origin along the Z axis, the vertical angle (vA) of the direction the cone is facing, and a horizontal angle (hA) along with ...
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26 views

Completing sets of numbers solely with trigonometric functions and an initial zero?

Last week an extra-curricular math academy I attend gave us this question as a challenge: You start with $0$, and the only functions you can do are $\sin, \cos, \tan, \sin^{-1}, \cos^{-1}, \tan^{-1}$ ...