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).

In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles.

Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.

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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 use trigonometry to create nice smooth curves.…
Starkers
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Why do complex numbers lend themselves to rotation?

In the introductory complex analysis course I am taking, nearly every theorem relates to rotation and argument. Why do complex numbers love doing this so much? I can understand why these theorems work; however, aside from basic knowledge of polar…
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How to find an angle in range(0, 360) between 2 vectors?

I know that the common approach in order to find an angle is to calculate the dot product between 2 vectors and then calculate arcus cos of it. But in this solution I can get an angle only in the range(0, 180) degrees. What would be the proper way…
Savail
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What is the advantage of measuring an angle in radian(s)?

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.
MrAP
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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, the minimum value of (or a lower bound…
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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 that its area is an even integer. I think…
space
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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: $$e^{i\theta}=\cos\theta+i\sin\theta$$ This enables us to…
KKZiomek
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Why do we say that the inner product "captures" the notion of "angle"?

It is well known that, via the polarization identity, a norm (which captures the notion of length) uniquely specifies an inner product; equivalently, if two inner products induce the same norm then they are the same inner product. My question: If…
EE18
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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 increase their value FROM LEFT TO RIGHT AND TOP TO…
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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 Hour = 360° 1 minutes = 6° Total Minutes covered by…
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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?
ymfoi
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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 parts?
AHB
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Finding a missing angle in the picture containing regular hexagon and square

I want to find $\angle AGM=\theta$ in the following picture: Here $ABCDEF$ and $BAGH$ are regular hexagon and square respectively and $M$ is the midpoint of $FH$. I found a trigonometric solution. I'm providing key ideas of the solution: Let…
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Find the angles of given triangle ABC

A triangle $ABC$ with angle bisectors $AA_1$ and $BB_1$ is given, such that $\angle AA_1B_1 = 24^\circ$ and $\angle BB_1A_1 = 18^\circ$. Find the angles of the triangle. I've been stuck on this one for quite a long time. After denoting with $I$…
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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 set of algebraic angles has some pretty neat…
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