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Questions tagged [angle]

An object formed by two rays joining at a common point, or a measure of rotation. In the latter form, it is commonly in degrees or radians. Please do not use this tag just because an angle is involved in the question/attempt; use it for questions where the main concern is about angles. This tag can also be used alongside (geometry).

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118
votes
9answers
14k views

Why is $\cos (90)=-0.4$ in WebGL?

I'm a graphical artist who is completely out of my depth on this site. However, I'm dabbling in WebGL (3D software for internet browsers) and trying to animate a bouncing ball. Apparently we can ...
25
votes
10answers
6k views

What is the advantage of measuring an angle in radian(s)? [duplicate]

What is the advantage and use of measuring an angle is radian(s) compared to degree(s)? My book suddenly switched to radian(s) for measuring an angle in this grade and I do not know why.
21
votes
1answer
677 views

Maximum angle between a vector $x$ and its linear transformation $A x$

Let $A \in \mathbb{R}^{n \times n}$ be a given symmetric positive definite matrix. I would like to find the maximal rotation $A$ can create over any unit vector $x \in \mathbb{R}^n$. In other words, ...
18
votes
3answers
2k views

proving that the area of a 2016 sided polygon is an even integer

Let $P$ be a $2016$ sided polygon with all its adjacent sides perpendicular to each other, i.e., all its internal angles are either $90$°or $270$°. If the lengths of its sides are odd integers, prove ...
16
votes
2answers
6k views

Is there a way to draw a 1 degree angle using only ruler and compass?

There are ways to draw $180^\circ, 90^\circ, 45^\circ, 30^\circ, 60^\circ, \dots$ angles. But is there a way to draw a $1^\circ$ angle? In other words how to divide a circle into $360$ equal ...
14
votes
6answers
3k views

What makes radians superior to turns/revolutions?

1. THE CONTEXT OF THE PROBLEM This question came to me when I was exploring complex exponents. The key identity to computing expressions with complex exponents is the Euler's identity: $$e^{i\theta}=...
14
votes
3answers
575 views

Have “algebraic angles” been studied before?

I'm writing a geometric software library and I came up with a useful concept. Let's call a real number $\alpha$ an algebraic angle if $\alpha\in[0,2\pi)$ and $\cos \alpha$ is an algebraic number. The ...
14
votes
2answers
468 views

BdMO 2016 National Secondary Problem 3.

$\triangle{ABC}$ is isosceles with $AB=AC$, $P$ is a point inside $\triangle {ABC}$ such that $\angle{BCP} = 30^{o}$, $\angle{APB} = 150 ^{o}$, $\angle{CAP}=39^{o}$. Find $\angle{BAP}$ This ...
12
votes
6answers
3k views

Angle between the hour and minute hands at 6:05

What is the angle between the hour and minute hands of a clock at 6:05? I have tried this Hour Hand: 12 hour = 360° 1 hr = 30° Total Hour above the Clock is $\frac{73}2$ hours In Minute Hand: 1 ...
12
votes
6answers
1k views

Why in calculus the angles are measured in radians? [duplicate]

Why is the formula $\lim\limits_{h\rightarrow 0}\frac{\sin h}{h}=1$ not valid when $h$ is measured in degrees?
10
votes
6answers
8k views

Given two vectors, how can I denote the angle between them?

Given the vectors $\vec{a}$ and $\vec{b}$, how can I denote the angle between them?
10
votes
3answers
2k views

Why are radians dimensionless? [duplicate]

According to https://en.wikipedia.org/wiki/Dimensionless_quantity, "A dimensionless quantity is a quantity to which no physical dimension is applicable." The article then explains, a few sentences ...
9
votes
7answers
1k views

What is a simple (or not) way of finding the lengths of the diagonals of this rhombus?

I recently taught a group of geometry students about properties of rhombuses, and gave them a set of homework exercises created by a previous instructor which included the following problem. If a ...
9
votes
3answers
4k views

Determine angle $x$ using only elementary geometry

Using only elementary geometry, determine angle x. You may not use trigonometry, such as sines and cosines, the law of sines, the law of cosines, etc.
9
votes
1answer
45k views

Find the bearing angle between two points in a 2D space

I continue developing a 2D Collision Detection System in a programming language (Javascript) and one of the last things I need to sharpen it is to know a formula to find this angle: NOTE: X and Y ...
9
votes
4answers
5k views

“World's Hardest Easy Geometry Problem”

This question is a "corollary" (if you will) to the World's Hardest Easy Geometry Problem (external website). Formally, this is called Langley's Problem. The objective of that problem was to solve for ...
9
votes
3answers
391 views

Making a regular tetrahedron out of concrete

I'm trying to make the following tetrahedron made of concrete just for fun: Each edge is a beam with a triangular cross section. I imagine the easiest way is to make 6 identical truncated triangular ...
9
votes
5answers
571 views

Given a circle of radius r, and two points ('X' and 'Z') on that circle, can some circumcircular arc “XYZ” be constructed of length r?

I am strictly an amateur, not a professional mathematician or some such. This question occurred to me while considering the fact that an angle of 1 radian centered on the center of a circle will ...
9
votes
1answer
215 views

Prove that two angles are equal

$M$ is the midpoint of $BC$ in the triangle $\Delta ABC$. $D$ lies on $AC$, and $AD = BD$. $E$ lies on the line $AM$, $DE$ is parallel to $AB$. How can I prove that the angles $D\hat{B}E$ and $A\hat{C}...
8
votes
3answers
13k views

Construct 60° angle through point, other line in only four compass-and-straightedge steps

PROBLEM Here is a surprisingly intriguing challenge posed on Euclidea, a mobile app for Euclidean constructions: Construct a 60° angle through both a point $P$ and an external (infinite) line $\...
8
votes
3answers
148 views

Angle Between Two Vectors Facing A Point

I need a mathematical algorithm for finding the angle, formed by three points, which is open toward a fourth point. For example, in Fig1 below I desire angle $\theta$ because it is "facing" point $P$...
8
votes
1answer
85 views

Internal angles in regular 18-gon

This (seemingly simple) problem is driving me nuts. Find angle $\alpha$ shown in the following regular 18-gon. It was easy to find the angle between pink diagonals ($60^\circ$). And I was able to ...
7
votes
2answers
148 views

Can there be two adjacent solid angles?

Thanks for reading. My real question is the second part - in the first part I'm just explaining myself. Please read through! Thanks. In 2D geometry, it is easy to picture what it means to add up 2 ...
6
votes
2answers
3k views

How many reflex angles can a polygon have? [closed]

I think the first of the polygons that can have a reflex angle is the pentagon. For a hexagon, a maximum of 2 reflex angles is possible. I tried to draw many concave polygons to find out a ...
6
votes
4answers
269 views

Length of segment parallel to an edge

I've tried all the possible side splitter and angle bisector theorem stuff and I still can't come up with the correct answer. I even tried some law of cosine and sine stuff, but nothing. Any help ...
6
votes
2answers
138 views

Is it possible to find such an angle using only angle chasing?

I've been trying to solve some problem and I came down to the following seemingly easy question: given two triangles ABC and ABD, and their corresponding angles, how do we find the angle $\angle ACD$ ...
6
votes
1answer
350 views

Name and number of “equilateral tessellations with same angles on all vertexes”

Longer background, shorter questions below: Tessellations of 2D plane consisting of regular polygons are usually described with vertex configurations such as "3.4.6.4" meaning that there are a ...
6
votes
1answer
140 views

How significant is the difference between averaging angles and averaging unit vectors?

I understand that numerically averaging angles (which I will call the $shortcut$ average for convenience) is in general going to produce a very different result than converting to cartesian unit ...
6
votes
1answer
84 views

Do angle properties (e.g. interior angles of a triangle sum to $\pi$) hold in general inner product spaces?

I've been rediscovering inner product spaces recently and have developed a couple of questions about angles defined in general inner product spaces. Consider the real vector space $V$ equipped with ...
6
votes
1answer
77 views

Are there two non-congruent quadrilaterals with same sets of sides and angles?

Are there two non-congruent quadrilaterals with same sets of sides and angles? It is relatively easy to construct such pentagons. Trying to construct quadrilaterals allows for seemingly multiple ...
5
votes
3answers
866 views

Product Identity Multiple Angle or $\sin(nx)=2^{n-1}\prod_{k=0}^{n-1}\sin\left(\frac{k\pi}n+x\right)$ [duplicate]

I have run across this interesting identity that I am unable to verify. $$\sin(nx)=2^{n-1}\prod_{k=0}^{n-1}\sin\left(\frac{k\pi}n+x\right)$$ Can anyone provide a hint as how one would prove this? ...
5
votes
4answers
287 views

Integrating with respect to an angle [duplicate]

Hello maths community! One day I was solving a geometry problem and I thought I had found a way of solving it. When I was solving the problem, I kind of invented a new way of finding an area of a ...
5
votes
3answers
254 views

$\tan(a) = 3/4$ and $\tan (b) = 5/12$, what is $\cos(a+b)$

It is known that $$\tan(a) = \frac{3}{4}, \:\:\: \tan(b) = \frac{5}{12} $$ with $a,b < \frac{\pi}{2}$. What is $\cos(a+b)$? Attempt : $$ \cos(a+b) = \cos(a) \cos(b) - \sin(a) \sin(b) $$ And we ...
5
votes
1answer
166 views

Calculate bevel edge Icosphere

I have an Icosphere with 80 faces, 120 edges. Now i am looking to find out what the angle is of the bevel between all the faces. With the bevel i mean the following see the image below: So i am ...
5
votes
2answers
748 views

Name of an angle between 0 and 180 degrees / $\pi$

Is there a name for and angle larger than 0 and smaller than $\pi$ or 180 degrees? So it covers acute, right and obtuse, it's kind of opposite of reflex angle. In my language there is a name for such ...
5
votes
1answer
96 views

Why only two dihedral angles for a snub dodecahedron?

The Wikipedia page for the snub dodecahedron provides explicit coordinates for its vertices, from which one can of course by brute force calculate that there are only two dihedral angles. Can anyone ...
5
votes
1answer
707 views

Quaternions: Why is the angle $\frac{\theta}{2}$? [duplicate]

The equation for creating a quaternion from an axis-angle representation is $$x'= x \sin\left(\frac \theta 2\right)$$ $$y' = y \sin\left(\frac \theta 2\right)$$ $$z' = z \sin\left(\frac \theta 2\right)...
5
votes
3answers
182 views

Solve for the angle x?

The problem seems really impossible to me, despite solving many problems on triangles. So I hope that you can solve it. src:https://www.instagram.com/p/BtUNj3IBIW5/
5
votes
2answers
132 views

Angle formed by summing $n$ unit vectors

I'm interested in the angle formed by the sum of $n$ unit vectors. Said angle must be a function of the angles of the $n$ unit vectors. Specifically, suppose that the $i$-th unit vector's angle is $\...
5
votes
1answer
79 views

Axioms for angular or conformal structure

Let $V$ be a real vector space. Is there a way to (directly) axiomatise the notion of a map $\Theta: V \times V \to \mathbb{R}$ being a measure of the angles between vectors? If we have an inner ...
5
votes
1answer
1k views

Help calculating angles for woodworking

I am working on a wood working project and need to cut some 2 x 2's on an angle in order to form an X inscribed inside a rectangle. Visually here is what I am trying to create: So basically I want to ...
5
votes
0answers
76 views

Dynamics of Triangle Iterates by angle bisector

I'm attempting to prove what is demonstrated in this Wolfram demo Let $T_0$ be an arbitrary triangle with vertices $A_0,B_0$and $C_0$,and let $T_1$ be the triangle formed by the intersection points ...
4
votes
3answers
7k views

Prove that a triangle is right triangle

I'd like to know if there's any theorem to prove that the triangle ACB' is a right triangle and that the angle ACB' is 90°. We know that ACB and A'B'C' are right triangles, so in my opinion ACB' is ...
4
votes
2answers
161 views

If $\sin(x)=\sin(\pi/4 + x)$, then why isn't $x=x+\pi/4$?

I've been solving a question, If $\cos(x) + \sin(x)=\sqrt{2} \cos(\pi/2 - x)$ then find the value of $x$. We know that $\cos(x) + \sin(x)= \sqrt{2} \sin(\pi/4 + x)$. So, $$\sin(\pi/4 + x) = \cos(...
4
votes
5answers
188 views

Show that the angles satisfy $x+y=z$

How can I show that $x+y=z$ in the figure without using trigonometry? I have tried to solve it with analytic geometry, but it doesn't work out for me.
4
votes
2answers
213 views

Prove these two angles are equal

$ACB=ACD$ $BAC=ADC$ $MB=MA$ $Prove \angle MCB=\angle MBD$
4
votes
2answers
1k views

How can I find the radius of curvature of a pipe when given the angle?

I am given a pipe with a 3mm diameter with walls of nearly infinite thinness (so the impact is not affected the the thickness of the pipe) that has something travelling down the center of it in a line....
4
votes
1answer
137 views

Find angle inside isosceles triangle

Please help. I have tried to find the answer but I couldn't. I have tried to draw a picture and measure the angle. It gives the answer $\theta = 25^\circ$ but I don't know how to do it. Thank you
4
votes
3answers
29k views

How do you find the sine of the angle between two vectors?

I do not know what the sine of the angle between two vectors is. I think it may be the vector created by connecting the tips of the two vectors but I am not sure. How do you find the sine of the ...
4
votes
4answers
109 views

Find angles in a triangle, with two similar triangles with scale factor $\sqrt{3}$

Triangle ABC has point D on BC which creates triangle ABD and ACD. They differ with the scale factor $\sqrt{3}$. What are the angles? I know ADB and ADC cannot be right, as it shares the side AD and ...