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-7
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2answers
31 views

Double angle sine [on hold]

Solve the equation: $$\sin2x=\frac 1{\sqrt{2}},\quad \text{for } 0< x <\pi$$
4
votes
1answer
43 views

Standing at the center of a cube and walking halfway to a wall - field of vision

In my python programming class one of the bonus problems is this: Suppose you are located at the exact center of a cube. If you could look all around you in every direction, each wall of the cube ...
0
votes
1answer
41 views

Do the $2^n$ hyper-octants of a $n$-sphere always have a $n$-dimensional right angle? Is $\pi/2$ only fundamental in $2$ and $3$ dimensions?

In $2$ dimensions, a $2$-sphere can be divided into $2^2 = 4$ congruent pieces, the $4$ quadrants, each of angle $\pi/2$ radians. In $3$ dimensions, a $3$-sphere can be divided into $2^3 = 8$ ...
0
votes
0answers
16 views

How are vertical angles always congruent?

Vertical angles are formed on opposite sides of two intersecting lines/line segments. I've also seen them as congruent angles because they have the same degree measurement, but how does this happen? ...
1
vote
2answers
30 views

Finding the angle of $\angle x$ and $ \angle y $

In the figure below, $ABCD$ is a rhombus and $ADE$ is a straight line. $\angle DAB = 51°$ and $\angle DCE = 42°$. what is the value of $x$ and $y$?
1
vote
1answer
336 views

How to find coordinates of 3rd vertex of a right angles triangle when everything else is known?

I want to locate precisely the 3rd coordinate of a right angled triangle. I have: the length of three sides The three angles The other two coordinates of the triangle The triangle can lie in any ...
0
votes
1answer
128 views

Solid Angle Integration

Can somebody explain the equivalence between integrating over the surface of a unit sphere and integrating over solid angle? I have been trying to understand the following transformation using a ...
1
vote
0answers
23 views

Solid angle subtended in latitude-longitude maps

I need to scale a latitude-longitude map with the solid-angle each "pixel" subtend. How can I obtain the said solid angle starting from the $\phi$ and $\theta$ angles? Thank you very much
1
vote
2answers
462 views

Solving for Cos Exactly

How to solve $\cos(\dfrac{5\pi}{6})$ and $\cos(\dfrac{7\pi}{6})$ exactly? I couldn't use special triangles to solve this either.
1
vote
1answer
32 views

Is the Inverse of the Vectorised Solid Angle Equation for $n$ Circular Discs Continuous?

I have a continuous function$^{*1}$ that takes in 3 arguments, and returns 24 outputs. I want to know if the inverse of this function is continuous. The 3 input arguments are the x, y, and z position ...
1
vote
1answer
40 views

Generalisation of the concept of solid angles in curved space

I wonder if there are standard extensions of the concept of solid angle in curved space. There should be because physicists who study radiation in curved space would run into this kind of needs. If ...
1
vote
1answer
167 views

Calculating solid angles

For a project, I'm working on calculating solid angles using calculus, but when I test my formula for the solid angle of a regular tetrahedron, I end up with $0.4633$ when according to Wolfram, I ...
0
votes
0answers
90 views

Solid angle of an off axis disk

In the diagram on the first page of this paper, what does the point c represent? Alternatively, what does the length R, the length r, or the angle theta represent? This is what I know so far. The ...
1
vote
0answers
229 views

Proof of solid angle theorem

I have a homework problem to prove about the solid angle. The book says: Let S be a smooth parametric surface and let P be a point such that each line that starts at P intersects S at most once. ...
0
votes
2answers
54 views

Calculate continous points on a circle circumference

Known details Circle radius r Origin (x,y) starting point in circumference say (u,v) I ...
0
votes
1answer
137 views

Needing Further Explanation of Knudsen's Cosine LAw

I'm reading over a paper by R. Feres and G. Yablonsky titled Knudsen's Cosine Law and Random Billiards, and I can't get around how they don't show directly how Knudsen's Cosine Law was derived. I'm ...
4
votes
1answer
224 views

Online Math Open Contest 2 Problem 50

In tetrahedron $SABC$, the circumcircles of faces $SAB$, $SBC$, and $SCA$ each have radius $108$. The inscribed sphere of $SABC$, centered at $I$, has radius $35.$ Additionally, $SI = 125$. Let $R$ be ...
0
votes
1answer
398 views

how to derive relation between solid angle and surface area and the radius of sphere using definite integral?

how to derive relation between solid angle and surface area and the radius of sphere ? I know $s=r^2\Omega$ but how they got it using integral ?
1
vote
1answer
91 views

Instrinsic definition of concave and convex polyhedron

Is it possible to distinguish a concave polyhedron from a convex one by mesurements made only on its surface, without a reference to the 3d space around it?
0
votes
1answer
107 views

Transport theory basics: can't understand solid angles

I don't understand something in transport theory: $$P(x,\vec{w})=p(x,\vec{w}) \cos\theta \, dw \, dA$$ This is the number of particles flowing across a differential surface element in the direction ...
2
votes
1answer
416 views

Calculate angle of triangle

I need to calculate the angle between two sides, I have the length of A & B sides, but don't know how to find the angle... Both sides are the same length. I can get the start and end vectors of ...
3
votes
1answer
278 views

A tricky integral (flux of a point charge through a disk)

The integrals: $$ \oint \frac{r\,dr\,d\phi}{\left(L^2+r^2+h^2+2Lr\cos\phi\right)^{3/2}}\\ \oint \frac{dx\,dy}{\left((L+x)^2+y^2+h^2\right)^{3/2}} $$ If we have a point charge at the origin and we ...
0
votes
1answer
69 views

Solid angle definition - can it be seen shown using an image?

How do we show on a picture that a solid angle equals to this equation i found on a Wikipedia: \begin{align} \Omega =\!\!\!\int\limits_{\vartheta ...
2
votes
1answer
485 views

Angle between different rays (3d line segments) and computing their angular relationships

I have several positions (say A,B,C,..) and I know their coordinates (3d). From each point, if a certain ray is passing in a way to converge them at a given (known) point (say O). This point O ...
1
vote
2answers
521 views

Solid angle and projections

During my studying of physics, I've been introduced to a concept of a solid angle. I think that I do understand it pretty good, however, I'm stuck with one certain problem. We know that a solid angle ...
3
votes
0answers
145 views

Calculating the Solid Angle

$\textbf{Problem:}$ Consider we have a class $C^1$ parameterization $\psi:[a_1,b_1]\times[a_2,b_2]\rightarrow\mathbb{R}^3-\{0\}$ for the surface $S$. Also, consider that $S$ is such that the map ...
1
vote
1answer
37 views

Finding the angle.

first question here. I ran into a problem where my math skills are just not enough, probably it's simple, but I don't know how to approach it. I'll show you this graphic so you can understand what I ...
0
votes
1answer
99 views

Given an angle at a point A and given another segment BC, construct a point D so that the angle DBC equals the given angle at A.

I'm not sure if I understand this question right. Q. Given an angle at a point A and given another segment BC, construct a point D so that the angle DBC equals the given angle at A. Figure - ...
0
votes
1answer
180 views

Explain solid angle $\Omega=\int\int_S \frac{\bar r \cdot \hat n dS}{r^3}=\int\int_S \sin(\theta) d\theta d\varphi$

I am trying to understand this formula from Wikipedia here. It is a generalization of radian. I am trying to do unit check but I cannot see how the units match. Steradion must be dimensioless unit ...
0
votes
1answer
153 views

Relationship between angles in tetrahedron

Let's say I have a tetrahedron like this in image: Do angles $CAD$ and $CBD$ equals in general tetrahedron?
4
votes
4answers
3k views

Why is the differential solid angle have a $\sin\theta$ term in integration in spherical coordinates?

When you integrate in spherical coordinates, the differential element isn't just $ d\theta d\phi $. No. It's $\sin\theta d\theta d\phi$, where $\theta$ is the inclination angle and $\phi$ is the ...