Trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles, and other topics relating to measuring triangles.
12
votes
0answers
651 views
Computing ${\int\limits_{0}^{\chi}\int\limits_{0}^{\chi}\int\limits_{0}^{2\pi}\sqrt{1- \cdots} \, d\phi \, d\theta_1 \, d\theta_2}$?
This question led me to this integral I can not solve:
$$
...
8
votes
0answers
147 views
On Ramanujan's Question 359
In JIMS 4, p.78, Question 359 was asked by Ramanujan. If,
$\sin(x+y) = 2\sin\big(\tfrac{1}{2}(x-y)\big)\tag1$
$\sin(y+z) = 2\sin\big(\tfrac{1}{2}(y-z)\big)\tag2$
prove that,
$\big(\tfrac{1}{2}\sin ...
8
votes
0answers
179 views
Evaluating $\int{ \frac{\arctan\sqrt{n^{2}-1}}{\sqrt{n^{2}+n}}} dn$
How to integrate?
$$\int{ \frac{\arctan\sqrt{n^{2}-1}}{\sqrt{n^{2}+n}}} dn$$
I have no idea how to do it.
Tried to get some information from wiki, but its too hard :|
6
votes
0answers
472 views
Nasty Integral - Closed form solution?
Any suggestions on how to integrate this beast?:
$$\int_0^{\omega_t}\int_{\omega_t}^f\sin^2(\omega_{12}/2)\sin^2(\omega_{23}/2)d\omega_{23}d\omega_{12}$$
where:
$f = 2\pi+2\tan^{-1}(y,x)$
$y = ...
5
votes
0answers
105 views
integral of the product of a trigonometric and an exponential function
Since tan has an odd power I would normally aim to sub $u=\sec(x)$, but I cant get rid of the $2^x$.
$$\int 2^x \tan^9(x^2)\sec(x^2)dx$$
I also tried integrating by parts but it got more complicated. ...
4
votes
0answers
47 views
How to find the maximum diagonal length inside a dodecahedron
I am trying to find the maximum length of a diagonal inside a dodecahedron with a side length of 2.319914107*10^89 meters. I am not sure if any other information than that is needed, if it is I ...
4
votes
0answers
193 views
Find area bounded by two chords and an arc in a disc
Math people:
Let $D$ denote the unit disc in $\mathbb{R}^2$ with center $(0,0)$. Let $\theta \in (0, \pi)$, and let $U \subset D$ be the horizontally symmetric region obtained by removing from the ...
4
votes
0answers
85 views
Geometric definitions of hyperbolic functions
I've learned in school that all the trigonometric functions can be constructed geometrically in terms of a unit circle:
Can the hyperbolic functions be constructed geometrically as well? I know ...
4
votes
0answers
109 views
Integral of a gaussian function of trigonometric functions
I need help with the analytical solution of this integral:
...
4
votes
0answers
81 views
Rational multiples of $\pi/2$ whose sines are also rational
Let $f(x)=\sin(x\frac{\pi}{2})$.
Let $R$ the set of $x$ such that :
$0\le x\le 1$
$x \in \mathbb Q$
$f(x) \in \mathbb Q$
Hence, $0\in R$ as $f(0)=0$. $1\in R$ as f(1)=1. And $\frac{1}{3}\in R$ as ...
4
votes
0answers
131 views
Reference for a tangent squared sum identity
Can anyone help me find a formal reference for the following identity about the summation of squared tangent function:
$$
\sum_{k=1}^m\tan^2\frac{k\pi}{2m+1} = 2m^2+m,\quad m\in\mathbb{N}^+.
$$
I ...
4
votes
0answers
175 views
Calculating equidistant points around an ellipse arc
As an extension to this question on equiangular fisheye distortion, how can I calculate equidistant points around an ellipse (or 1/4 segment of) given it's aspect ratio?
When it's circular, I can use ...
4
votes
0answers
166 views
Reprojecting/converting an orthographic image/grid into a cartesian grid
I'm trying to dewarp a fisheye image into a simple rectilinear image of a subset of the fisheye. As part of this, I'm trying to map the azimuth/altitude values into a point on the image.
The points ...
4
votes
0answers
187 views
$\arctan$ of constant multiplied $\tan \varphi$
Helllo, I'm new here. I will try this great place to get my answer. I have a next problem:
$$\vartheta = \arctan(K \tan \varphi)$$
$K > 0$
Where $K$ is a constant. If $K = 1$ then $\vartheta = ...
4
votes
0answers
244 views
Find roots of sum of sinusoids
Given this function and an initial point, find the next root:
$$
\begin{align}
f(t) & = -L\\ & {} + A \sin(\Theta_1 + \omega_1 t) \\ & {} +B \cos(\Theta_1 + \omega_1 t)\\ & {} - ...
4
votes
0answers
77 views
how understand if a segment is inside a lissajous curve
i am a programmer and not a math guru, but i like geometry. so if i'm not accurate in math terminology or i have folly question please sorry me.
i'm drawing with a programming language the lissajous ...
3
votes
0answers
57 views
looking for reference or nice proof of trig lemma
Math people:
I am looking for a reference or a nice proof of the following fact. I have proven it myself, but my proof is messy: let $\theta \in (0,1]$ and $\alpha \in (0, \frac{1}{2}\theta^2]$. ...
3
votes
0answers
244 views
Trigonometric inequality proof
Can anyone help me in proving that $$\cos\theta > \frac{\left(x^a\cos\theta-(x-1\right)^a\cos\frac{\ln x\theta}{\ln(x-1)})\cos(\theta+\gamma)}{\cos\gamma},$$ where $a<1$, $x\in \mathbb{N}$, and ...
3
votes
0answers
136 views
Prove that if $x$ is a non-zero rational number, then $\tan(x)$ is not a rational number and use this to prove that $\pi$ is not a rational number.
Prove that if $x$ is a non-zero rational number, then $\tan(x)$ is not a rational number and use this to prove that $\pi$ is not a rational number.
I heard that this was proved two hundred years ...
3
votes
0answers
258 views
Proof of the sine rule
So I made my first attempt at a proof. I think it turned out well. Maybe not. But I was wondering if someone could take a look at it and tell me what they think. I'd be glad to hear some criticism on ...
3
votes
0answers
158 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
0answers
46 views
secants, exponentials, quotient structures
Trigonometric functions are . . . . . . somewhat like exponential functions.
If $f$ is an exponential function, then $\displaystyle \frac{\prod_i ...
3
votes
0answers
94 views
Solving a linear trigonometric equation
Let $n$ be a natural number. For $a_i,\omega_i,\varphi_i \in \mathbb{R}$ how can one find solutions $x \in \mathbb{R}$ for the equation:
$$\sum_{i=1}^n a_i \cos( \omega_i \cdot (x-\varphi_i)) = 0$$
3
votes
0answers
74 views
How do I calculate the size of padded envelope needed for a given size box?
When you put a box of dimensions (W x H X D) in a padded envelope of dimensions (X x Y), what is the mathematics?
The padded envelope also has to have a flap of length (F), it also has welded seams ...
2
votes
0answers
35 views
Bernoulli generating function and cotangent
May I ask for a little help in solving a problem about Bernoulli number generating function?
Bernoulli number generating function is given by: $$f(z):=\begin{cases}
\frac{z}{e^{z}-1} & z \in ...
2
votes
0answers
106 views
Solving the equation $\cos(x) \cdot \cosh (x) + 1 = 0$
$$\cos(x) \cdot \cosh (x) + 1 = 0$$
Sorry I am a software developer and I have forgotten this part of mathematics! What is the value of $x$ in the above equation?
I need the steps to solve the ...
2
votes
0answers
158 views
Dangerous substitutions in finding trigonometric integrals.
This is a question NOT on how to actually solve the problem but more about a concern in what one needs to know before making a sort of substitution.
Last day I tried to solve this integral: $\int ...
2
votes
0answers
59 views
Desired Z axis and Yaw to ZXY Euler Angles?
I'm trying to calculate a desired pair of pitch and roll Euler angles (the XY in ZXY format) given a desired z-axis of the rotated frame (expressed in the world frame) and a specified yaw angle ...
2
votes
0answers
47 views
Rationality of Polynomial Coefficients. Integral Question.
We are entertaining polynomials with roots, all unique, on the curve $\Upsilon_s = \{{( 1-\cos[\theta])^{-s} \exp(i \theta)} \ | \ \theta \in \mathbb{R} \}$, where $s>0.$ $\Upsilon_s$ looks like a ...
2
votes
0answers
46 views
tangents of infinite sums — reference request
This section of Wikipedia's List of trigonometric identities was, as far as I know, written entirely by me. For a time, it dealt only with finite sums, and said something like this:
$$
...
2
votes
0answers
86 views
flaw in law of cosines usage
I'm going to through a list of coordinates and computing the angle between every two adjacent lines. In other words, I'm computing an angle for every 3 consecutive points. Every three consecutive ...
2
votes
0answers
123 views
Product of sines
I am looking to evaluate
$$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n}$$
without using complex numbers. I can show the result if $n$ is a power of $2$, but if $n$ is anything else I reach a point where I ...
2
votes
0answers
101 views
Inequality with even powers of trigonometric functions
For $m>0$,
$0 < n\leqslant m+1$ ($m,n\in \mathbb{Z} $) , and $0 < a < 1$ , prove that
$$2^{n}\cdot \left( a^{n}\cos ^{2m}\dfrac {\pi a} {2}+\left( 1-a\right) ^{n}\sin ^{2m}\dfrac {\pi ...
2
votes
0answers
67 views
References for “closed form” numeric solutions of $\tan x=-a x$
I am looking for references that discuss solutions of the equation
$\tan x=-a x$
(for $x,a\in \mathbb{R}$). I know about the graphical approaches, and any number of numerical solution approaches, ...
2
votes
0answers
65 views
Functional inverse of $(a + b\sin\theta)^2\tan\theta$
So, I revisited to the situation involved in my previous question, with the intent of generalizing it to any two masses and charges. When I started going through the model again, beginning with the ...
2
votes
0answers
95 views
How can this trigonometrics equation be solved exactly, if possible?
I was working on an approximation for the sine function, in which I needed to calculate the maximum error to work on a compensation polynomial. My approximation was this:
$$f(x) = \frac {4} {\pi^2} x ...
2
votes
0answers
117 views
Sine rule and equal angles
Is it true that if a triangle on a unit sphere has 2 sides with equal length then their opposit angles must be equal? I think it is true. I think we can use the spherical sine law. Call the sides with ...
2
votes
0answers
760 views
how to find mid point of an arc?
I have start point $(x_1,y_1)$ and an end point $(x_2,y_2)$ and radius of arc. How to calculate the co-ordinates of mid-poing of arc? The arc is the part of a circle.
Known Values
...
2
votes
0answers
125 views
A trigonometric inequality involving sine
Let $0<a<\pi/2,0<b<\pi/2$, $0<\lambda<1, \mu=1-\lambda$. Does anyone see a good proof
of the inequality:
$$\sin(\lambda a)\sin(\lambda b)+\sin(\lambda a)\sin(\mu ...
2
votes
0answers
145 views
Maximum size of a rotated-then-cropped rectangle
With regard to topic/question New size of a rotated-then-cropped rectangle:
The answer by Isaac,
the maximum area is $b^2\csc\alpha\sec\alpha$ when $x=0.5b\csc\alpha = 0.5b/\sin\alpha$
seems to ...
2
votes
0answers
167 views
Geometrical interpretation of trigonometric antiderivative
I know about geometrical explanation of [defininte] integral as an area under the curve, and I wonder if there are some ideas, which may give similar insight in taking antiderivatives [indefinite ...
1
vote
0answers
36 views
Basic Trigonometric Substitution Question
I have a basic trig substitution question for integrals. It always seems that x is opposite to the theta angle. However, making x the adjacent on the right angle triangle seems to work just fine as ...
1
vote
0answers
35 views
simplification of a natural log of a trigonometric function
hope you are all well.
I am having a bit of a mental block, I am wondering if it is possible to simplify the following expression:
$$k\cos X \cdot 4\ln(\cos X)$$
where $k$ is a constant and $X$ ...
1
vote
0answers
24 views
Can you help me reverse the Minimum Curvature Method?
The minimum curvature method is used in oil drilling to calculate positional data from directional data. A survey is a reading at a certain depth down the borehole that contains measured depth, ...
1
vote
0answers
46 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 ...
1
vote
0answers
28 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?
1
vote
0answers
23 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 = ...
1
vote
0answers
49 views
At large times, $\sin(\omega t)$ tends to zero?
While doing a calculation in quantum mechanics, I got a expression $\sin(\omega t)$, and my prof said if I consider the consider at large times, then i can assume that this goes to zero because at ...
1
vote
0answers
74 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 ...
1
vote
0answers
44 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 ...
