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

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84 views

3d Implicit Trigonometry help?

I'm trying to understand implicit 3D trigonometry, specifically with this equation: $$\sin(y)+\cos(z)=\cos(x)$$ Can someone please explain to me what is going on with this equation? I really can't ...
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122 views

Euler's Basel problem continued… $\zeta(2n)$ expressed in terms of $sinc$?

I have to make a brief intro before comming to my question. To approach the famous Basel problem Euler starts with the $sinc$ function \begin{align}\frac{\sin(x)}{x} = 1 - \frac{x^2}{3!} + ...
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101 views

Is there a continuous version of $tan^{-1}(\frac{y}{x})$ for the entire unit circle?

The fact that $tan^{-1}(\frac{y}{x})$ only "works" for the upper-right quadrant makes some calculations (for a physics simulator) impossible. I of course use $atan2(y,x)$ in the code, that's not what ...
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94 views

simplification of a natural log of a trigonometric function

hope you are all well. I am having a bit of a mental block, I am wondering if it is possible to simplify the following expression: $$k\cos X \cdot 4\ln(\cos X)$$ where $k$ is a constant and $X$ ...
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268 views

Solution procedure for a system of trigonometric equations in two variables

i would like to know if there's a method for solving the following system using (or not) tan half angle substitution. $$A\cdot\sin(\theta_1) + B\cdot\cos(\theta_1) + C\cdot\sin(\theta_3) + ...
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111 views

Strange results for solving boundary angle of full reflection

I was trying to solve the following group of equations for $\alpha$ using Wolfram|Alpha: $$\frac{v_2\cdot\sin(\alpha)}{v_1} = 1 \\ v_1 > 0 \\ v_2 > 0$$ I expected something like $\alpha = ...
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64 views

How can I align the angle between points with the magnetic heading as the points move?

I have 3 robots which must track a point. The distance between all the robots and the point is known so a triangle can be formed between any 2 robots and the point. If I find the angles in the ...
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349 views

How can I find the compact trigonometric Fourier series from these signals?

I've been stuck on this for a while, but how exactly would I go about calculating the compact trigonometric Fourier series for both of these signals? I have a general formula down for it, but I just ...
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33 views

Why is there a difference which way you rotate in different systems?

At school I've always rotate clockwise, with $90^0$ being straight up. But I see some system operates with $0$ being straight up and clockwise rotation, does someone know why this is?
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73 views

Pure Phase Number

I am read a solution (4.9) Here say: ... both $a, d$ are pure phases, so that it is always possible to find (non unique) real numbers $\alpha, \beta, \delta$ such that $a = ...
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132 views

Shortest time taken for a targeted object with a set speed to meet a body orbiting in a circle

I'm trying to figure out how to find the optimum point that a ship in 2D space would meet a planet which was orbiting in a perfect circle. The orbit is at a constant rate, and the ship can only move ...
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130 views

Differential geometry textbook or lecture notes on the riccati equation and riccati inequality

I took a course on differential geometry and didn't get one specific topic well, so I am searching on some additional metrial to understand it in a better way. This wasn't a course about classical ...
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221 views

How does trigonometry in a Galois field work?

This is a follow-up to this question. I'm interested in doing trigonometry in finite fields on a computer. I do not understand precisely how trigonometric functions are supposed to work in a finite ...
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80 views

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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315 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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189 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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167 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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441 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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77 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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124 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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230 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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175 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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274 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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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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243 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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4 views

Multiplicity in Solutions to Trig Function Equations

This is a very simple problem, but I can't figure out where I am going wrong! Say you have the following: $a \sin\theta + b \cos\theta = c. \tag{1}$ Now, this for example can be rewritten using: $R ...
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10 views

What do I need to know before Trig?

I finished a summer course that covered Math 1 Honors and I want to enter Trigonometry/Algebra 2 this year. What do I need to know before entering this class (I haven't taken Geometry)?!
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26 views

Expected value of product of sinusoids

In the book Adaptive Signal Processing by Widrow, an equation (2.20) on page 23 is presented without proof as: $$E \left[ x_k x_{k-n} \right] = \frac{1}{N} \sum_{k=1} ^{N} \sin\left(\frac{2 ...
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12 views

Single-statement Continuous Periodic function without trigonometry and complex numbers

Superseding the question Periodic function without trigonometry and complex numbers , I am now asking: Can I get a single-statement continuous periodic function without using trigonometric functions ...
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28 views

Epsilon-Delta Limit for Trigonometric Function

I'm studying an Epsilon-Delta proof for a trigonometric function: $$\lim_{x \to 1/9} \sin(x) = \sin(1/9)$$ This is the procedure from my (Italian) book: $$−\epsilon < \sin(x) − \sin(1/9) < ...
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25 views

Get coordinates to rotate a path around a circle JS (d3.js)

I'm trying to use the formula from this question Calculating the coordinates of a point on a circles circumference from the radius, an origin and the arc between the points to rotate a line around 180 ...
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15 views

Trigonometric identities for Bessel Functions?

I'm wondering if there exists extensions of trigonometric identities to special functions like Bessel? For example, is there an alternative way to express the following? $J_0((a+b)x) = ?$ Thanks
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38 views

Simplify addition of 4 sin()s into one term

Is it possible to turn $\sin(a)+\sin(b)+\sin(c)+\sin(d)$ into one term, such that there is no addition or subtraction? I've tried by using $$ \sin(a) + \sin(b) = 2 \sin \left(\frac{a+b}{2} \right)\cos ...
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20 views

Trigonometric functions on set of co-primes to $n$

E.g. I plotted the function values of $y=\tan(x\frac{\pi}{n})$ for integer $x$-values in the range of $0,\cdots,n$ where $n$ is a given odd integer ( to prevent the case of undefined ...
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48 views

If $A=\frac{\pi}{7},$ Then $\tan A\cdot \tan 2A+\tan 2A\cdot \tan 4A+\tan A\cdot \tan 4A$

The value of $\displaystyle \tan \left(\frac{\pi}{7}\right)\cdot \tan \left(\frac{2\pi}{7}\right)+\tan \left(\frac{2\pi}{7}\right)\cdot \tan \left(\frac{4\pi}{7}\right)+\tan ...
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40 views

How to find values of non linear equations / system and solve for given values

I'm trying to find the value for the variable phase in a equation / system if amp=0.5 and freq=2.5 (note: i'm looking for several different phase values given amp and freq but this is a small ...
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15 views

Slopes from differentiation of logit function are too high that angle calculation seems off

I'm differentiating the logit function to plot a relationship between x and y: $f(x) = 1/1+exp(-X\beta)$ with the differentiation to find the slope of the above equation: $f(x)' = \beta ...
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27 views

Prove that $r^3\rho=2R \rho \rho_1\rho_2 \rho_3$

I doubt whether this question is correct or not. Because in the LHS, it is fourth dimension in length and in the RHS, it is fifth dimension in length. If correct, I don't know how to prove it. If ...
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19 views

Right Triangles With Altitudes To Hypotenuse

Create 2 similar right triangles, with hypotenuse 3.00 & 4.00, whose Sin of each acute angle is 0.60 and 0.80. NOTE: The 2 triangles, when joined correctly, will form the 3 4 5 right triangle ...
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26 views

Arc Length for Superposition of Sinusoidal Curves

I am wanting to compute the arc length, $s$, of a superposition of two sinusoidal functions--say $$y(x) = A\cos\left(k_1 x\right)+B\cos\left(k_2 x\right).$$ There is a special relationship between ...
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23 views

Smoothly interpolating between functions to create a bouncing wave

How can I create a function which allows me to control the roundness of a wave so I can transition between an Round Wave -> Linear Wave -> Inverted Round wave? I've made a function which creates a ...