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

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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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176 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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163 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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430 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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119 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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219 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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172 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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265 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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239 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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20 views

can these two trig functions cross?

I have two functions; $$f(i) = \frac{1+A^2 \cos{\frac{2 \pi i}{N}}}{1+A^2}$$ and $$G(j,k) = \frac{(1+A^2 \cos{\frac{2 \pi j}{N}})(1+A^2 \cos{\frac{2 \pi k}{N}})}{(1+A^2 ) ^2}$$ and ...
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31 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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9 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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24 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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11 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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39 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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36 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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38 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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27 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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20 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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6 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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38 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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29 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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43 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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10 views

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 $\frac{m(m-1)}{2}$ ...
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21 views

How to determine the period of the following functions?

How would the following make a difference to the period of a function? $$ \cos(t)~~~~ (1)$$ $$ \cos(\omega t)~~~~ (2)$$ $$ \cos(\omega t + \phi)~~~~ (3)$$ Would this be right, $(1)$ has period ...
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44 views

As the Moon revolves around the Earth, the side of the Moon that faces the E..

As the Moon revolves around the Earth, the side of the Moon that faces the Earth is partially illuminated by the Sun. These "phases" of the moon are described by a fraction F of a lunar disc. For ...
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58 views

How to calculate a person's Latitudinal and Longitudinal location based off of Sun and time

INTRO I remembered hearing about it being possible to calculate a person's position or the position a picture was taken, based on time of day and the position of the Sun, position meaning latitudinal ...
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44 views

Is there such thing as a Co-theta?

I plan to make a notation for trigonometry called co$\theta$ An example is that co$\theta$ is equal to 45 and $\theta$ is equal to 45. Co$\theta$+$\theta$=90. In this equation, ...
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44 views

Fourier Expressions

In the Fourier series, what are all the ways we can express: $\displaystyle\sin\left(\frac{n\cdot\pi}2\right)$ $\displaystyle\cos(n\cdot\pi)$ I know we can express as $(-1)^{(n+1)}$, and as ...
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29 views

Coordinates of a point between two other points with a exact distance from one of them

In this situation, I want the coordinates of the point C, I already know the coordinates of A and B, and the distance D (the distance between A and C). C must be on the A-B segment.
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23 views

Weierstrass Trig Substitution Proof

After browsing some topics here, through one post, I discovered the "miraculous" Weierstrass substitutions. However, I can not find a decent or "simple" proof to follow. The simplest proof I found is ...
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34 views

Asymptotic analysis of Integrals of powers of sine and their application to intersections of hyperspheres

I am trying to estimate the probability of an event in an algorithm. For simplicity, assume there are two hyperspheres of radius $r$, at a distance $r$ from each other. I am looking to see how the ...
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30 views

How to approximate a trigonometric to make less computation complexity

I having a trigonometric function such as $$ p_2(s) = \begin{cases} \frac {1}{(2 \pi)^2}(1-\cos (2 \pi s)), & \text{if $s \le1$ } \\ \frac {1}{2 }(s-1)^2, & \text{if $s >1$ } ...
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22 views

Sine and cosine graph transformation

I'm having some difficulties with this question A bike is on a stand such that the highest point of the back wheel is 47 inches above the ground. If the pedal is turned counter clockwise, the back ...
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12 views

Why do I get these remarkable wave patterns when I put in the sine of (the coefficient of except 0) for x: y=sin((coefficient barring zero)x)

I saw this inequality grapher on Math Is Fun and when I was playing around with it one time, I got something so remarkable: when you enter the sine of (coefficient barring zero)x, it gives you a wave ...
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20 views

finding the formula for a sinusoid given 3 points

I have a sinusoid function, but I don't know the frequency or amplitude. I have 3 distinct points which I know lay on the sinusoid. Is this enough information to derive the sinusoid function? ...
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44 views

Why does the graph of $y=\cos 2x+\cos 4x$ look so different from the functions $y=\cos 2x$ and $y=\cos 4x$?

What is $y=\cos 2x+\cos 4x$? and why does its graph look so different from the functions $y=\cos 2x$ and $y=\cos 4x$? I don't understand why this function does not follow the general rule ...
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8 views

Change transform matrix for a figure in a sequence?

Here is what I want to do All the drawn figures are rects but each one has its own transform matrix which includes translate, rotate, scale and perspective transforms. I want to transform the left ...
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20 views

8 Angles Question: What are solutions with all angles rational multiples of pi?

I don't know how to draw a picture, maybe someone can help. Consider a convex quadrilateral $ABCD$. The 8 angles I'm referring to are the angles made between the diagonals and the edges. Explicitly: ...
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25 views

How fast is this dot moving when the angle $θ$ between the beam and the line through the searchlight perpendicular to the wall is $π/6$?

A searchlight rotates at a rate of $4$ revolutions per minute. The beam hits a wall located $11$ miles away and produces a dot of light that moves horizontally along the wall. How fast (in miles per ...
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32 views

Dirichlet integral using real Analysis

The teacher made this approach to solve the Dirichlet integral , $$ J_n= \int_0^\frac{\pi}{2} \frac{\sin(2nx)}{\sin x}\:\mathrm{d}x,\quad I_n = \int_0^\frac{\pi}{2} \frac{\sin(2n+1)x}{\sin ...