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

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Let triangles $ABC$ and $DEF$ be inscribed in the same circle. If the triangles are of equal perimeter,then prove that …

Let triangles $ABC$ and $DEF$ be inscribed in the same circle. If the triangles are of equal perimeter,then prove that $$\sin A+\sin B+\sin C=\sin D+\sin E+\sin F$$ Also state and prove the converse ...
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26 views

How are the sine functions along with the hyperbolic functions visualized with imaginary rotations?

Since we know that: cos(t)=cosh(it) and isin(t)=sinh(it) I've been thinking about this, and obviously this is referring to how if you move at a right angle from a circle on a conic section, you end ...
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45 views

integral 2D involving complex exponential and cosine

I've some doubts about my solution of this integral: $$I(\phi_{1},\phi_{2})=\int_0^ {2\pi} \,d\phi_{1}\int_0^ {2\pi} \,d\phi_{2} \frac{e^{-in\phi_{1}} e^{-im\phi_{2}}}{2\pi}\frac{e^{il\phi_{1}} ...
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42 views

Is there a function to standardize the resultant when a point vector is reflected by a cone?

In a previous question Reflect a point vector in a conical surface and determine average resultant vector. an expression:- $V1_x = P_x (1 - 2\sin^2(\gamma))$ $V1_y = - P_y (1 -\sin^2(\gamma))$ ...
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237 views

How do I solve this bearing/direction problem?

At 2:00 PM, a ship leaves port and travels N15degreesE at a rate of 20 mph. At 2:30 PM, another ship leaves the same port and travels S75degreesW at 30 mph. How far apart are the two ships at 4:30 PM? ...
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36 views

Finding identities for $\cos(a+\frac{\pi}{4} \mod \frac{\pi}{2})$

The question is rather self-explanatory, but I can't find good answers online. The reason for wanting identities for this is that $b$ and $c$ are constants, and this formula is being used on a device ...
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59 views

Solving physics problems using real and imaginary numbers

I was working on a particular physics problem and like we usually in physics do - replaced $\cos(x)$ with $e^{ix}$ and worked the result. When I tried to solve it without complex numbers I stuck. In ...
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49 views

Does $a x+b=\cos(x)$ have a special-functions solution analogous to the Lambert W function?

The Lambert W function is defined as the solution to the equation $z=w e^w$, in the sense that for all $z\in\mathbb C\setminus(-\infty,-1/e]$ we can find a complex number $W(z)$ which obeys ...
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20 views

Where does this formula for prediction of a multiple wave come from?

On the slide I am reading, it says this: Any surface related multiple can be construced of primaries: $\hspace{5cm}$Multiple = Primary$_1$ (green) + Primary $_2$ (red) $$T_2 = T_1 + ...
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135 views

Directional derivative - angle between the vector and coordinate axes

Doing an exercise a about directional derivatives, it was required to find the derivative of a given function $f(x,y,z)$ in the direction of the vector $ \vec{v}$ that forms with the coordinates axes ...
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33 views

Finding the coordinates of the top and bottom circles of a moving and rotating cylinder in 3D

I have a cylinder that is moving and rotating in a 3D space. I need to calculate the coordinates of the center of the cylinder's top and bottom circles. Here's the information I have : I have at the ...
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41 views

Re-arranging Pythagoras

The background: I have two accelerometers mounted at right-angles to each other. Lets call them $A_x$ and $A_y$. They give a stream of integer numbers over the network representing acceleration, ...
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297 views

Matrix with trig functions and Cramer's rule

Using Cramer's rule solve for $x'$ and $y'$ in term of $x$ and $y$ $x = x'\cos\theta - y'\sin\theta\\ y = x'\sin\theta + y'\cos\theta$ So what I have is this $\det\begin{bmatrix} \cos\theta& ...
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30 views

Help solving a function

I have $2$ equations and $2$ angels that i need $V_{1}$ and $V_{2}$. I know The Point $(X_{m}, Y_{m})$ and the point $(X_{a}, Y_{a})$. I have one point $(X_{p}, Y_{p})$ that moves with the equation ...
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77 views

Help me debunk mathematically a driving law.

I was wondering if you guys can assist me calculating the relative probability of an accident scenario given the following info. Disclaimer: What we will look at now is just a "model" or a ...
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20 views

Resolving bearings for tilted structure (vector rotations)

I am trying to find a neat solution to resolving bearings for a tilted structure. For example, if I know that a hub is $268.74^{\circ}$ and offset ($X = -3.542 $m, $Y = -1.857$ m, $Z = 2.013$m) with ...
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61 views

Linearising angle to chord length over a reasonable domain of < pi; or, how to make measuring a rock with a protractor easy

Imagine that I have a protractor and compass, and wish to use it to measure the distance between two points (potentially in three dimensional space, such as on a rock). However also being a forgetful ...
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395 views

Sum of Sinusoids with Same Frequency = Sinusoid (proof)

I am studying Fourier analysis on my own, I realised that probably the first thing you want to proof in Fourier transform is that the sum of 2 sinuoids (namely a sine and cosine) with the same ...
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170 views

Converting a 3x3 matrix to euler angles in various rotation orders

So let's say I have a 3x3 orientation matrix set up like: a b c d e f g h i which I have generated from euler angles in a left-handed, Y-vertical system. ...
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44 views

Triangle ABC. 3rd point.

I have a triangle ABC, xyz coordinates of points A and B are known. Also, length AB, AC and BC are known. How can I get the xyz coordinate of point C with respect to the xyz frame?
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26 views

Find vector with simple trigonometry

I've spent too much time solving a fluid mechanics problem because of this trigonometry. How do I find $V_{t2}$ ? The answer is $rw - V_{n2} \cot(\theta)$. (can't seem to get equations to work)
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86 views

Product of Sinusoids: Sum of Frequencies (proof)

In this article: http://en.wikipedia.org/wiki/Negative_frequency#Applications they write: "the product of two complex sinusoids is also a complex sinusoid whose frequency is the sum of the original ...
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20 views

Is it possible to find an exact expression for system for recursion of non linear equations?

I have a system of N non-linear equations of N unknowns $\phi_i, i \in 1,2,... N$. By specifying the middle two unknowns as $\phi_{N/2} = 0, \phi_{N/2+1} = X$, I get that: \begin{align*} ...
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16 views

Parametrically defined Spheres in $R^n$

So I have 2 questions here which are closely linked: How do you parametrically define the circle $(x')^2 + (y')^2 = r^2$ using (x') and (y') as coordinates on the plane ax + by + cz = 0 that are ...
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185 views

Law of cosines on cyclic quadrilateral

‎Consider‎ the ‎triangle ‎$\triangle‎‎ ABC$ ‎with‎ ‎area‎ $n$. Let $a$ denote the lenght of side $A$; and similarly with $b$ and $B$; $c$ and $C$. Denote ‎$\theta$‎‎ as the measure of angle ...
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91 views

How to Find End Point, after rotation

I am having an 3D object, length of the object is 27.5 meter, rotation value is -30 degree and the rotate origin point will be one end. After rotating the object i want to find the coordinate of ...
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108 views

Trying to find coordinates of another point using bearings

I am trying to help one of my siblings with a trig project and there is one part that I am having a little trouble with. In his project, he is trying to find the coordinates of a UFO given the ...
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71 views

Prove the Lobachevsky-Bolyai formula for the Klein model

I want to prove that e^(-d) = tan(Π(d)/d) in the Beltrami-Klein Model for the angle of parallelism in correspondence to the distance d, where d is the klein distance d(AB) = (1/2)|ln((AB,PQ)). A hint ...
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69 views

Get the entrance point from a straight line in a rectangle

The rectangle is like a street. The right half is to go upwards, the left half to go down. The red lines are paths of vehicles. And my goal is to give every vehicle the right lane. So when you look at ...
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105 views

Getting the angle between three points

So I have this psuedo code here (converted from c# to show you better) ...
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82 views

What is this expression called?

Could anyone please tell me if they recognize this equation? What it does is calculate the angle between two lines, but I need it's name. Any help is greatly appreciated! $$\sin \theta = A_{1} \cdot ...
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80 views

Any obvious simplification of sin(x) / sin(y)

I have an expression that is made up of the sines of several angles unfortunately none of them are "friendly" angles such as 60 or 45 but sine occurs several times. I get the feeling there might be ...
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923 views

General solution of trigonometric function

For any real numbers $x$ and $y$. $\sin x = \sin y \implies n\pi + (-1)^ny, $ where $n \in Z$ If $\sin x = \sin y$, then $\sin x -\sin y = 0$ or $2\cos\frac{x+y}{2}\sin \frac{x-y}{2} = 0$ ...
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74 views

Finding anti derivative

It is mentioned in a different thread that $U(x)=\sin\left(\dfrac1{\ln(1+x^2)}\right)$ is an elementary function. My question is, how do you integrate it then?
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74 views

Torus equation in terms of tangent

So if I have an equation for a torus in $F(a,b) = (X, Y, Z)$ where $X = (R + r\cos a)\cos b$ and $0 < r < R$, how would I go about rewriting this equation for $X$ in terms of $\tan(a/2)$ and ...
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109 views

A construction of the trig functions on the unit circle

Can anyone shed some light on this picture? http://upload.wikimedia.org/wikipedia/commons/9/9d/Circle-trig6.svg I am not interested in "$\sin$", "$\cos$", or the outdated trig functions. How do we ...
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40 views

Solving for $R$ given $\tan p=\frac{18H}{243-H^2}$ and $R(243-H^2)\cos p+18HR\sin p=1$

Ok so we start with $$\tan p=\frac{18H}{243-H^2}$$ And use this in the equation $$R(243-H^2)\cos p+18HR\sin p=1$$ To find $R$ in terms of $H$ without trig functions I have the answer by the way, ...
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75 views

Trigonometry Question

In trigonometry to measure the height or distance of objects we consider the distance between the observer and object to be straight. But the surface of the earth is curved. Assuming the line to be ...
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61 views

Rearrange $y = \frac{\tan\left(\frac{N x}{2}\right)}{N}$ to give N

Is it possible to rearrange $$y = \frac{ \tan \left(\frac{N x}{2}\right)}{N}$$ where $x \lt \pi$ as a function of x and y that gives N?
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98 views

Reverse Engineering (Inverse?) complex trigonomic function

So, I have this nifty function: ...
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134 views

Determining a point in 3D space

So given a point, a rotation around the y-axis, a rotation around the x-axis, and a distance, how can one calculate the relative point in space? For example, the beginning coordinates are (0,0,0). ...
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92 views

$\cot(x)$ or $\tan(x)$ amplitude with $F(x)$ or $G(x)$?

If you are doing $f(x)$ and $g(x)$ of a tangent/cotangent function and you get an amplitude. Should you write the final equation with or without the amplitude because technically tan and cot don't ...
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268 views

Distance between two objects in a picture

lets say I have a photo that has a picture on a wall and a book upright on the desk. now i know the size of both of these objects. I want to find the distance between two of them on the photo, I was ...
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127 views

Calculate distances with Lat and Long

I am trying to calculate a waypoint a set distance away from my current location. I know my Lat and Long of my current location and the distance away a want the waypoint to be. I also know the ...
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197 views

Calculate coordinates of the a point in space with hypotenuse and two angles given

I have a cylinder with a length of $2$, and two angles for rotation around two of the axes. Functions for that are named $\text{RotX}$ (rotation around X axis) and $\text{RotZ}$ (rotation around Z ...
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41 views

best way to detect the trigonometric identites that shall work on a given expression so as to simplify it accordingly?

how to tell that what trigonometric identity (a.k.a. Pythagorean trigonometric identity) will work on the given equation , so then you can simplify the equation accordingly in order to apply that ...
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86 views

Approximating a function with a sine function: transform into constant amplitude?

I have a smooth function, it is stationary. So I tried approximating my function with regression by fitting a sine function that changes period, phase & frequency every observation to get the ...
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145 views

Solve this trigonometric system $ \tan x+\tan y=2\sqrt3 \land \tan\frac{x}2+\tan\frac{y}2=\frac{2\sqrt3}3 $

$$ \tan x+\tan y=2\sqrt3 \land \tan\frac{x}2+\tan\frac{y}2=\frac{2\sqrt3}3 $$ I need full solution please. I've tried different transformations, but couldn't get much near, I keep getting huge ...
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111 views

Set of all points which are a specified angle away from a given point on a sphere.

I have a sphere with a known point on the surface in polar coordinates. I'm looking to find the set of all points which are exactly some angle away from this point in polar form (this should describe ...
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181 views

Double integration involving polynomial functions and sinc function

I encountered a problem which I can't seem to simplify/solve. I was wondering if any mathematicians or specialists knows how to approach this problem? $$\int^{0.5}_{-0.5} \int^{0.5}_{-0.5} \; ...