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

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How were trigonometrical functions and its inverses discovered?

Imagine you just did a circle. Some functions are just definitions, like $\sin$,$\cos$ and $\tan$ but how do you derive a formula to get the $\sin$ from an angle in radians (maybe by Taylor series, ...
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307 views

Unit-circle versus trigonometric approach to introducing Calculus

I'm trying to decide on a text for a refresher in pre-Calculus. The publisher of two books by the same set of authors differentiates between them as follows: Precalculus, Seventh Edition This ...
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66 views

If $\sqrt{2}\cdot \cos A = \cos B-\cos^3 B$ and $\sqrt{2}\cdot \sin A = \sin B-\sin^3 B$, determine $\sin(A-B)$

If $\sqrt{2}\cdot \cos A = \cos B-\cos^3 B$ and $\sqrt{2}\cdot \sin A = \sin B-\sin^3 B$, determine $\sin(A-B)$ My Try: $\sqrt{2}\cdot (\cos A-\sin A) = (\cos B-\sin B)-(\cos B-\sin ...
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117 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 http://upload.wikimedia.org/wikipedia/commons/9/9d/Circle-trig6.svg I am not interested in "$\sin$", "$\cos$", or the outdated trig functions.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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133 views

Some trigonometric equation problems

show that : $$\left(1+\cos \frac{2\pi}{13}\right)\left(1-\cos \frac{4\pi}{13}\right)\left(1+\cos \frac{6\pi}{13}\right)\left(1+\cos \frac{8\pi}{13}\right)\left(1-\cos ...
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183 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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164 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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438 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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76 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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60 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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69 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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87 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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121 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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222 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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82 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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174 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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267 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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2k 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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5k 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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2k views

Compute Altitude and Azimuth using either Quaternions or Rotation Matrix or Roll, Pitch and Yaw component

I am struck with a mathematical problem. I want to convert the iPhone device's attitude information which is available in one of the following forms: Quaternion Rotation Matrix Roll, Pitch and Yaw ...
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240 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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12 views

Find Intersection of Two Circle given Lat/Lon and radius

I am attempting to calculate the intersection of two circle on the Earth with a given latitude, longitude and radius. I started with this post. While I am using this in the context of programming, it ...
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48 views

Zeta function, how to solve a finite geomatry summation.

I wanted to solve the zeta function for an undifend period "$d$". So for every $d\ge2$. $$\zeta(-s)= \frac{1}{(d^{s+1}-1)}\sum_{m=1}^{\infty} \frac{1}{2^{m+1}}\sum^{m}_{j=1} ...
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26 views

Rotating one coordinate system about another

I have two coordinate systems: A and B. I also have a point p, whose position relative to ...
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77 views

Is there a space in which the $\vec a$ in $\sin(a_1\cdot x)+\sin(a_2\cdot x)$ is linear?

I have equations of the form $\sin(a_1\cdot x)+\sin(a_2\cdot x)=y$ (actually more complicated, but that's the general essence). I want to solve for $\vec a$ using linear regression instead of ...
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20 views

Calculating point following with rotation

as my question my sound about programming it's really just a math. I just want to know how to calculate it not write it in programming language. So, I want to create effect, which looks like this: ...
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29 views

Trigonometric Substitution Method to solve Cubic Equation.

Here are the questions. IN the wiki page, it says p has to be smaller than 0. But they didnt really explain why... Therefore, I assume it is impossible to have a complex number inside arcosine, is ...
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24 views

Characteristic polynomials for matrix A, involving the Identity matrix

Let us say we have a square matrix A, where A's characteristic polynomial is defined as $P_A(t) = \det (t I - A)$ (In this problem, I represents the identity matrix which has the same dimensions as ...
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48 views

Need to solve for an angle, do I need to use numerical methods??

I need to run a simulation on MATLAB where I need to solve two equations for two unknowns. The equations look something like this, $$x_{comp} = G\cdot \cos^2 b \cdot \cos(a+b).$$ $$z_{comp} = ...
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26 views

Get the coordinate offsets with known rotation angles (i.e. Yaw, Pitch, Roll)

I'm working on correcting an tilted object to its vertically placed position. Below is my drawing illustrating my situation: http://i.stack.imgur.com/0XotT.png Assuming: I have a stick stood on a ...
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37 views

how do you find the distance between 2 points with known distances between other points

link to diagram for explanation: http://i.imgur.com/8cmuWib.png I am trying to determine the distance between i and j. These nodes are all placed in a coordinate system. things I know: ...
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31 views

At what angle does the stone need to be hit?

In curling, it is often necessary to hit and displace an opponent’s stone to win the end. Olivia would like to hit her opponent’s stone with her own stone. If she releases her stone at the hog line, ...
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164 views

Solve the equation (very hard)

How to find all irrational solutions of the equation ...
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43 views

Is it possible to define an inverse of the main three trig. functions without domain restrictions?

Ok, I know that the main three main trigonometric functions, that is the tangent, sine, and cosine, are periodic and thus not one-to-one, but onto. And, since an inverse requires a function to be onto ...
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22 views

Find marginal distribution (Integral Solution)

I have derived bivariate exponential distribution in term of polar coordinate system. Now I need to derive marginal distribution of $f(\theta)$ from joint $f(r,\theta)$ for this we have to eliminate ...
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40 views

Determine the minimum and maximum angles, to the nearest tenth of a degree, that a pipe can make with the horizontal.

For residential drains, a horizontal pipe needs to have a minimum slope of 1/4 inch per foot and a maximum slope of 1/2 inch per foot for waste to drain properly. This means that for every horizontal ...
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11 views

What does 3D gaze direction contains? And how to convert it to yaw and pitch?

I am trying to use a dataset. But I am facing two problems or confusions in understanding it. Can anbody guide me what 3D gaze direction stands for or means (angles, (x, y, z) coordinates or what)? ...
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27 views

initial height = 60“. There is 5 degree decline over 163.5”. What is ending height?

I'm building a roof for a structure and need to get the ending height correct. The initial height is 60". The adjacent length (the ground) is 163.5". The decline is 5 degrees. I have gotten the ...
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12 views

Composite trapezoid rule and trigonometric functions

I am trying to solve the problem talked about in: Trapezoid rule over trigonometric polynomials Show that the composite trapezoid rule over an equidistant partitioning with interval size ...
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23 views

Solve for 'y' for elipse rotated at an angle

How solve for y if we have set of x coordinates for elipse rotated at an angle 'A' ,has the origin at (h,k) and height as a and b
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43 views

Simplify trigonometric equation

$$ \alpha sin\theta + \beta sin\phi + \gamma sin(\theta+\phi) = 0 $$ where $\alpha, \beta, \gamma$ are constants. I want to simplify this into a linear relation between $\theta, \phi$ I wonder if ...
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38 views

Finding coordinates of position given three coordinates and distances: 3D

I'm hoping to determine the x, y, z coordinates of a 4th position (D) given the coordinates of three other positions and their distances: $A(0.25, 0.25, 0.25), B(0.4663, 0, 0.25)$, and $C (0.3912, ...
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40 views

Transform complex trigonometric expression with $\arccos$

In the proof that the poles of a Chebyshev filter lie on an ellipse, there is the following transformation, for the $s$ values correspondant to the poles. From (1) $$s_{pm} = j \cos \left[ ...
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31 views

Perspective view and calculating based on it

I've got a project in which i'm asked to use an image captured in perspective view of a lane road (Assume the distance of the camera and the angle relative to the road are known). What I need to do ...
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22 views

Computing nearest point in a cone of angle

Figure 1 In the above figure, there are 3 agents namely i_1, i_2 and i_3. For each agent, ...
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16 views

Determining pitch and roll angles from the coordinates of a vector

I want to know, given the measurement of an accelerometer at rest (so not really an acceleration but a force per unit of mass) the inclination of this accelerometer, along the X and Y axis. So, In ...
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42 views

Trigo Study plan

In what order of topics is probably the most effective in learning trigonometry for starters... where should I first start? and steps in between to De Moivre's Theorem (which is the last topic)... ...
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34 views

Making a metric out of distance measure

I'm working with a pseudo-distance measure that is not a metric since it does not hold the triangle inequality. It is called Dynamic Time Warping. The problem is - I need to perform some projections, ...
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45 views

Solutions of trigonometric equation $a\sin(x) + b\cos(x) = n$

Is there a solution of the equation $a\sin(x) + b\cos(x) = n$ in rational numbers (i.e. $a,b,n,x$ are rational and positive) where $x$ is not of the form $90n^\circ$? (This question was also there on ...
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Finding common roots to a variable number of functions

I am trying to solve the following problem. Given $a\in\mathbb R^n$, $u\in\mathbb{R}^n$, $m\in\mathbb{N}^\star$, Find the/some common roots $(t_1,...,t_m)$ of the $\frac{m(m-1)}{2}$ ...