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

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5
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74 views

How to find inverse of $\sin(x) + \sin(2x) = y$?

I was wondering if there were any way to solve the equation $$\sin(x) + \sin(2x) = y$$ in terms of $x$. This of course would allow us to express the inverse for this function on $-\frac{\pi}{4}$ to ...
5
votes
1answer
231 views

Reference for a tangent squared sum identity

Can anyone help me find a formal reference for the following identity about the summation of squared tangent function: $$ \sum_{k=1}^m\tan^2\frac{k\pi}{2m+1} = 2m^2+m,\quad m\in\mathbb{N}^+. $$ I ...
4
votes
1answer
159 views

Radical expression for Cosine formulas

Is there nice radical expression for $$\cos\left(\frac{\pi}{2^k+1}\right)?$$ Example: $\cos\left(\dfrac{\pi}{5}\right)=\dfrac{\sqrt{5}+1}{4}$. Please provide some concrete examples. Also please ...
4
votes
1answer
82 views

Calculate the X,Y values of an ellipse

I guess am confused somewhere. I have the length(l) and breadth(b) of an ellipse enclosing rectangle. I know the center point and the angle(x) that the line makes with the center. I want to know the ...
4
votes
1answer
160 views

Find area bounded by two unequal chords and an arc in a disc

Math people: This question is a generalization of the one I posed at Find area bounded by two chords and an arc in a disc . Below is an image of a unit circle with center $O$. $\theta_1, \theta_2 ...
3
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1answer
42 views

Find Area of 3 Sector Circle, Variable center point

I have a Circle separated into 3 sectors. At start each sector has the same central angle, 120°. Therefore each sector should be taking up the same area. I want to be able to move the center point ...
3
votes
1answer
50 views

Finding the equation for a sinusoidal cycle/function given points.

We are given the population of a fictional animal at different years: $$\begin{array}{l|r} \textrm{Year} & \textrm{Population}\\\hline 1945 & 347,0000\\ 1955 & 76,000\\ 1965 & ...
3
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1answer
75 views

Proving a tough geometrical inequality, with equality in equilateral triangles.

For any triangle with sides $a ,b, c$ prove or disprove (1) and (2) : $$\sum_\mathrm{cyc} \frac{1}{\frac{(a+b)^2-c^2}{a^2}+1}\ge \frac34$$ Equality in (1) holds if and only if the triangle is ...
3
votes
1answer
25 views

Possible trig identity?

Is there a trigonometric identity for $\sin(ab)$? Thanks in advance! I can't find it anywhere. Bothering me a lot. For that matter, what about $\sin(a^{-1})$? Both of these for cosine, too, but if ...
3
votes
1answer
129 views

Is there active research in trigonometry?

One learns trigonometry in high school/secondary school and either forgets it if one continues onto a career less mathematical or, possibly. uses it extensively in their work, as do engineers and ...
3
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1answer
44 views

Trigonometric identity for $a\sin 2x + b\sin(x+\alpha )$

The following is a known trigonometric identity, $$a\sin x+b\sin(x+\alpha )=c\sin(x+\beta )\,$$ where $$c=\sqrt {a^2+b^2+2ab\cos \alpha },\,$$ and $$\beta =\arctan \left(\frac {b\sin \alpha ...
3
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1answer
46 views

Evaluate using a trigonometric substitution

The Integral is $$\int x^3\sqrt{9-x^2} dx$$ My method: $$x = 3\sin\theta$$ $$dx = 3\cos\theta d\theta$$ $$\sqrt{9-9\sin^2\theta} = 3\cos\theta$$ $$= \int 27\sin^3\theta (9\cos^2\theta) d\theta$$ ...
3
votes
1answer
65 views

Express $(1+\cos(x-1))^3$ as a trigonometric polynomial in x.

Express $(1+\cos(x-1))^3$ as a trigonometric polynomial in x. I keep doing this problem and somehow I keep messing up the constants, and it just jumbles up in my head. $$(1+\cos(x-1))^3$$ $$= ...
3
votes
1answer
68 views

Find $\lim_{n\to\infty}\frac{\sum_{k=1}^{n}\cos k+\sum_{k=1}^{n}\sin k}{\prod_{k=1}^{n}\cos k\sin k}$

Find the following limit: $$\lim_{n\to\infty}\frac{\sum_{k=1}^{n}\cos k+\sum_{k=1}^{n}\sin k}{\prod_{k=1}^{n}\cos k\sin k}$$ The numerator can be simplified by using Euler's formula and the sum of ...
3
votes
1answer
131 views

A Functional Differential Equation

I was having a play with some trig. identities and noticed the following: $\cos{x}=\frac{\sin{2x}}{2\sin{x}}$ Now, $\cos{x} = \frac{d}{dx}\sin{x}$ so I made the following analogous differential ...
3
votes
1answer
100 views

Largest scaled rotated rectangle inside rectangle

There are two rectangles: $r_1$ and $r_2$. $r_1$ is rotated $\theta$ and then uniformly scaled by a factor $k$ to exactly fit within $r_2$. I'm trying to find the value of $\theta$ that maximizes $k$, ...
3
votes
1answer
107 views

Trig question, can this be solved?

I was wondering is it possible to solve this without assuming that CAD=DAB. As I use the law of sines, trigonometry and have tried to apply law of cosines. However, I cannot see how you can solve ...
3
votes
1answer
253 views

Tricky integration by substitution.

I have to get this integral (EDIT: it should definitely be 1-x^2 in numerator) $$\int_{-1}^{1} \frac{ \sqrt{1-x^2}}{1+x^{2}} dx$$ into $$\int_{-\pi }^{\pi } \frac{1}{1+\cos^2\theta } \,d\theta - \pi$$ ...
3
votes
1answer
540 views

How to plot N points on the surface of a D-dimensional sphere roughly equidistant apart?

Let's say I have a D-dimensional sphere with a radius R. I want to plot N number of points evenly distributed (equidistant apart from each other) on the surface of the sphere. It doesn't matter where ...
2
votes
1answer
31 views

Proving that $\tan(x) = x$ has exactly one solution per interval $((n-\frac12)\pi, (n+\frac12)\pi)$

I want to prove that $\tan(x) = x$ has exactly one solution per interval $((n-\frac12)\pi, (n+\frac12))$. My attempt: $\tan(x)$ is $\pi$-harmonic, and has a range of $(-\infty, \infty)$ for each ...
0
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0answers
68 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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94 views

Getting the angle between three points

So I have this psuedo code here (converted from c# to show you better) ...
0
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0answers
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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0answers
78 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 ...
0
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0answers
867 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 $2\cos\frac{x+y}{2}\sin \frac{x-y}{2} = 0$ ...
0
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0answers
73 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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0answers
72 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 ...
0
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0answers
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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0answers
79 views

How would I go about solving this Euler's Equation problem, getting even and odd components?

I'm stuck on this question in my signals and systems class, the question asks to find the even and odd components of the equation. Now I know that $e^{jx} = \cos(x) + j\sin(x)$, however this ...
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0answers
39 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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0answers
73 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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0answers
60 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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0answers
95 views

Reverse Engineering (Inverse?) complex trigonomic function

So, I have this nifty function: ...
0
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0answers
131 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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0answers
90 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 ...
0
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0answers
267 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 ...
0
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0answers
125 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 ...
0
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0answers
189 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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0answers
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 ...
0
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0answers
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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0answers
143 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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0answers
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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0answers
110 views

Sufficient to show that $\sinh(\operatorname{arcsinh}(x))=x$ for arcsinh being the inverse of sinh?

I have to show that arcsinh is the inverse function to sinh. I checked that $\sinh(\operatorname{arcsinh}(x))=x$. Is that sufficient or do I also need to show that ...
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0answers
177 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} \; ...
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0answers
325 views

trigonometry and formula

I asked the same question some time ago, but it is closed. This time I will be clearer (I hope). If I have a right triangle $c ^ 2 = a ^ 2 + b ^ 2$, and $b > a$ (so $a$ is the shorter side and ...
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0answers
63 views

How can I convert lines intersecting a plane into a focused image?

I am writing a particle transport code. I would like to be able to obtain an image of my geometry when transporting photons given the following information: The photons are incident on a plane. For ...
0
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0answers
77 views

Counteract preceding rotations

In a situation where I have two axis adjacent back to back (let's say a robotic arm) I can sometimes perform two rotations ($R_1, R_2$) such that the resulting position and direction is unchanged. ...
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220 views

Trig identities for $A\sin^2(x)+\cos^2(x)$

Does anyone know of any useful trig identities for manipulating $A\sin^2(x)+\cos^2(x)$? The only thing I come up with is: $A\sin^2(x)+\cos^2(x)=\frac{1}{2}(1+A)+\frac{1}{2}(1-A)\cos(2x)$ I'm trying ...
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0answers
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 ...
-1
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27 views

finding the height when given the angle of elevation and depression

standing at the top of a 20m tower, the forrest slopes away from the base of the tower at an angle of depression of 15degrees. she identifies a tree and uses a clinometer to find the angle of ...