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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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How to express an angle between two angle bisectors in interior angles of a convex quadrilateral?

Given a convex quadrilateral $ABCD$, I would like to express the angle between angle bisector of internal and external angles of the opposite vertices in interior angles of $ABCD$. Here is the drawing ...
Rusurano's user avatar
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2D angle from known 3D angle. [closed]

I have $2$ vectors in $3$D space: $\hat{u}$ and $\hat{v}$. The angle between them is $\varphi$, so that $$ \hat{u}\cdot\hat{v} = u_{x}v_{x} + u_{y}v_{y} + u_{z}v_{z} = \cos\left(\varphi\right) $$ ...
Ido's user avatar
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0 votes
3 answers
55 views

How to prove opposite angle bisector theorem for convex quadrilaterals?

Let $ABCD$ be a convex quadrilateral with $BL$ and $DL$ be its angle bisectors. I want to know how to prove that the acute angle $\alpha$ between these bisectors is equal to $\frac{\left|\angle A - \...
Rusurano's user avatar
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1 vote
0 answers
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solution-verification | show that $[BN$ is the bisector of the angle $\angle VBC$

The problem Let $VABCD$ be a regular quadrilateral pyramid with peak $V$. The plane $\alpha$ contains the line $AB$ and cuts $CV$ and $DV$ at the point $M$, respectively $N$. If $P$ is the midpoint of ...
IONELA BUCIU's user avatar
-3 votes
1 answer
35 views

Find the value of the interior angles of a polygon [closed]

I have coordinates of points $(x, y)$. By connecting these points we get a polygon in which I have to get values of its internal angles. For example points = $[3,1], [3,3], [1,3], [3,5], [7,5], [7,1]$ ...
Paul's user avatar
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0 answers
33 views

lower bound of $\sum \cos^k(\theta_i-\theta_j)$

In this question, Lower bound of sum of cosine of angle difference, the lower bound $\sum_{i,j\in [n]}\cos^2(\theta_i-\theta_j)\geq \frac{n^2}{2}$ is given in the answer. I am wondering, what is the ...
chloe's user avatar
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2 votes
2 answers
147 views

Find the Cosine of the Angle Between the Plane $(MND)$ and the Plane $(ABC)$

The problem Consider the cube $ABCDA'B'C'D'$. Let $M$ be the middle of $AA'$ and $N$ the middle of $B'C'$. a) Determine the tangent of the angle between $AD'$ and $MN$ b) Find the cosine of the angle ...
IONELA BUCIU's user avatar
4 votes
2 answers
192 views

solution-verification | Calculate the sine of the angle between two side faces of a trunk

The problem Let the trunk be a regular quadrilateral pyramid $ABCDA'B'C'D'$ with the side of the large base of $8$ cm and the side of the small base of $4$ cm. The lateral faces are isosceles ...
IONELA BUCIU's user avatar
0 votes
1 answer
21 views

Determine the measure of the angle of the planes $(NBC)$ and $(ABC)$

the problem Let the triangle $ABC$ with $\angle A=30, \angle B=15, AC=2a$ and $M$ be the midpoint of $AB$. At point M we construct the perpendicular to the plane of the triangle on which we take point ...
IONELA BUCIU's user avatar
1 vote
0 answers
50 views

lower bounding $\Big\{\Big(\sum_{i\in[n]}\sum_{j\in[n]}\cos(\theta_i-\theta_j)\Big)^2-{\sum_{i\in[n]}(\sum_{j\in[n]}\cos(\theta_i-\theta_j))^2}\Big\}$

Consider $$S(\theta):=\Big\{\Big(\sum_{i\in[n]}\sum_{j\in[n]}\cos(\theta_i-\theta_j)\Big)^2-{\sum_{i\in[n]}(\sum_{j\in[n]}\cos(\theta_i-\theta_j))^2}\Big\}$$ Motivation. I am curious about the minimum ...
chloe's user avatar
  • 1,052
1 vote
3 answers
219 views

Parallel line equation

I want to incorporate 2 diagonal lines in a logo design. The lines have to be parallel to each other and have to be exactly 0.5 inches apart when measured perpendicular. The upper point of Line 1 has ...
Geo's user avatar
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1 answer
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computing elevation angle of a object on a circular fisheye image

I'd like to calculate the elevation angle of an object at any given coordinates (x,y) of the image that has been taken with a circular fisheye lens (185° of fov). By elevation angle, I mean the angle ...
Sygall's user avatar
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0 votes
1 answer
45 views

How to calculate the clockwise rotation (bearing) from 3 known coordinates and independent of the cartesian XY axes

Greetings Maths experts, I come to you once more looking for mathematical assistance to help me solve another challenge in my CAD software. Background Info: I'm trying to write a VBA macro which will ...
SmartSolid's user avatar
0 votes
0 answers
24 views

How do I prove that the angle between two 2d vectors depends of sign of dot product of two 2D.?

How would you prove that given two 2D vectors in the $\vec{v} = \begin{bmatrix} v_{1} \\ v_{2} \\ \end{bmatrix}$ and $\vec{u} = \begin{bmatrix} u_{1} \\ ...
Alpha2017's user avatar
-1 votes
1 answer
88 views

How do you solve for the side length of this square?

I came across this question which had 3 parts. The first 2 were about showing what sin(a) equals which I managed to get, but the third part was show that $x^4-56x^2+640=0$ and solve for $x$, but how ...
qwerteee's user avatar
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0 answers
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Geometric Inequality with Angle Bisectors in a Triangle

Given triangle ABC with angle ABC = 60°. AP is a bisector of angle BAC. AQ is a bisector of angle CAP. Prove that BC > 4PQ. So far, I've managed to express the equality of the products of sides ...
Tan's user avatar
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1 vote
1 answer
37 views

Invariant angle between focus and intersections of tangents to an ellipse

Let $A,B$ be two fixed points on the ellipse $\mathcal{E}$ and $X$ a movable point moving along the ellipse $\mathcal{E}$. Let $M,N$ be the intersections of the tangent to $\mathcal{E}$ passing ...
J P's user avatar
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1 vote
2 answers
56 views

How to calculate angle between vectors in a way that is invariant under translation?

Normally you just take the two vectors $\vec A $ and $\vec B $ and simply dot them, so $\vec A \cdot \vec B $ then divide by the product of there lengths and finally take the cosine inverse. But this ...
Zero's user avatar
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1 vote
0 answers
35 views

What is an angle? [duplicate]

I am a high school student and recently I started trigonometry and one question that comes to my mind every time is that "What is an angle?" I mean when we say that angle between two sides ...
Himanshu Singh Nirwan's user avatar
14 votes
5 answers
521 views

Let $ABCD$ be a convex quadrilateral. If the measure of the angles $A=90, C=96, D=78$ and $BC=2*AB$, then the measure of the angle $ABD$ is?

The problem Let $ABCD$ be a convex quadrilateral. If the measure of the angles $A=90°, C=96°, D=78°$ and $BC=2*AB$, then the measure of the angle $ABD$ is...? The idea As you can see I calculated ...
IONELA BUCIU's user avatar
2 votes
0 answers
29 views

A result concerning heights in an arbitrary triangle

I am trying to understand if the following claim is true or false (probably true?). The claim says that each of the three heights $\{AA_1,BB_1,CC_1\}$ bisects the relevant angle of the blue triangle. ...
user237522's user avatar
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1 vote
1 answer
193 views

solution-verification | compare to angles in a rectangular parallelipiped

the problem Let $ABCDA'B'C'D'$ be a rectangular parallelepiped with $AB=3a, BC=2a,AA'=6a, a>0$. We denote by $M$ and $N$ the means of $AA'$ and $CC'$ and by $P$ the intersection of the lines $BA'$ ...
IONELA BUCIU's user avatar
0 votes
1 answer
86 views

solution-verification| calculate a trigonometric function of the angle of the plane (ABC) with the plane $\alpha$

the problem The triangle ABC has vertex A in a plane $\alpha$ and is projected onto this plane according to the isosceles right triangle AMN, with $AM=MN=a\sqrt{2}$ M being the projection of B and N ...
IONELA BUCIU's user avatar
0 votes
0 answers
72 views

How to find the tangent when converting from 2D angles to Spherical to Cartesian coordinates?

I am making a spherical/ball-in-socket joint, and I want to limit the movement of the bodies relative to each other. The limit is defined as 2 angles $\alpha$ and $\beta$ which make a 2D rectangle. I ...
Liburia's user avatar
1 vote
1 answer
44 views

Find the smallest integer, $k>3$, such that the angle is on the y-axis

You have an angle $(5π)/6$ multiplied by $k$. So $(5π*k)/6.$ Find the smallest integer for $k$ that is greater than $3$ such that the angle is either $π/2$ or $(3π)/2$. Find the integer value of $k$. ...
risa's user avatar
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3 votes
1 answer
92 views

Does every spherical sector of a circle in $\mathrm{M}_n(\mathbb{C})$ contain an invertible matrix?

Some context : (Notations : For the rest of the post, I will identify $\mathrm{M}_n(\mathbb{C})$ with $\mathbb{C}^{n^2}$, and I will note $\mathcal{S}^{n^2-1}$ the unit sphere of $\mathbb{C}^{n^2}$) I ...
Timothe Schmidt's user avatar
3 votes
3 answers
60 views

How to find the angle of an ellipse's semi-major axis given a rectangle with known width and height, and tangent to the ellipse on 4 sides?

Given a rectangle that contains an ellipse, and the ellipse is tangent on all 4 sides, how can I find the angle and lengths of the semi-major and semi-minor axes? I'm mainly interested in getting the ...
Tororoi's user avatar
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1 vote
1 answer
58 views

Deriving the Unit Quaternion to Tait-Bryan Angles conversion.

Let me start by saying I have a working solution. But I just don't understand how to get there. I've followed the well-written paper Technical Concepts Orientation, Rotation, Velocity and Acceleration,...
Michael Marcin's user avatar
0 votes
0 answers
38 views

How do you define a triangle from its orthographic projections on the 3 axis

I've spent hours on this problem and i'm running out of idea. Say you have an angle in a 3D space made of the points BAC, where A = (0,0,0), B and C can be moved and lengths AB and AC are always equal ...
Usylom's user avatar
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2 votes
1 answer
99 views

How can I calculate the angles of a given orthographic camera perspective?

In Blender, I can set up an orthographic / isometric camera like so: With the X, Y, and Z rotation set to those values, a cube has the following properties: all sides are rendered at equal length, ...
TKoL's user avatar
  • 131
2 votes
1 answer
111 views

Reflecting point across bisector of acute angle leads to perpendicular elsewhere

Exercise. Let $\angle BAC$ be an acute angle. In the interior of $\angle BAC$ there is a point $P$ whose projections onto the sides $AB$ and $AC$ are precicely $B$ and $C$. Draw the bisector of the ...
Linear Christmas's user avatar
2 votes
0 answers
39 views

Can angles be defined by norms that are not induced by inner products?

Can angles be (well-)defined in a normed vector space where the parallelogram law does not hold? In other words, if the norm is not induced by an inner product in a normed space (say $L^1$ space), can ...
chaohuang's user avatar
  • 6,399
2 votes
1 answer
195 views

Show that the points $M, N, P, Q$ are coplanar and concyclic.

the question Consider the tetrahedron $ABCD$, with the perpendicular edge $AD$ on the $(BCD)$ plane. Let $X$ be some point of the edge $AD$ and the point $Y$ on $(AD)$ such that $m(∠Y CD) = m(∠DAB)$. ...
IONELA BUCIU's user avatar
4 votes
4 answers
429 views

Looking for alternative proofs of this statement about angles

This is the theorem to prove. Below is my proof that I consider rather long and complex. The given data is on this drawing: Construct $\angle DCE = \angle DCB$. The point $E$ on ray $CE$ is chosen in ...
Rusurano's user avatar
  • 848
7 votes
6 answers
422 views

Show that triangle $ABC$ is isosceles.

the question Consider the triangle $ABC$ and a point $M$ inside the triangle such that $\angle MAB = 10 ,\angle MAC = 40 ,\angle MCA = 30 $ and $\angle MBA = 20 $ . Show that triangle $ABC$ is ...
IONELA BUCIU's user avatar
0 votes
0 answers
20 views

Lengths of divided hypotenuse by bisector of right triangle of a triangle, that has circle inscribed in it

A circle is inscribed in a triangle ABC. Knowing that |AB| = 20cm, |AC| = 16cm and |BC| = 12cm, calculate the lengths of the segments into which the bisector of angle ACB divides side AB. So from the ...
Plk's user avatar
  • 3
1 vote
2 answers
83 views

Looking for a simple proof of this statement about angle bisectors

Theorem: Two angle bisectors make angles with the sides opposite to the internal angles they bisect. If we construct angle bisectors from two formed angles which are opposite to the same side of the ...
Rusurano's user avatar
  • 848
-2 votes
1 answer
44 views

Angle in a triangle

The angle of triangle ABC also belongs to triangle AHC. The problem states that $\tan (\angle A) = 5$ $\frac{CH}{AH} = 5$ or $\frac{BC}{AC} = 5$ or both.
Kqry's user avatar
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0 votes
1 answer
113 views

Would Dividing circles in 2520 degrees instead of 360 degrees give extra advantage? [duplicate]

I have two questions I am aware of the fact that historically babylonians used sexagecimal systems and it had mathematical advantage that the number $360^\circ$ has all divisors from $1$ to $10$ ...
Dheeraj Gujrathi's user avatar
0 votes
1 answer
37 views

Knowing that the area of a lateral face is equal to the area of the base, find the measure of the angle formed by the planes $(MBN)$ and $(ABC)$.

the question Let $VABCD$ be a regular quadrilateral pyramid, $M$ the midpoint of the edge $VC$, $N$ the midpoint of the edge $AD$. Knowing that the area of a lateral face is equal to the area of the ...
IONELA BUCIU's user avatar
1 vote
1 answer
110 views

The volume of a tetrahedron $ABCD$ is $\frac{1}{6}$. Determine $CD$ knowing that $\angle ACB=45$ and that $AD+BC+\frac{AC \sqrt{2}}{2}=3$.

the question The volume of a tetrahedron $ABCD$ is $\frac{1}{6}$. Determine $CD$ knowing that $\angle ACB=45°$ and that $AD+BC+\frac{AC \sqrt{2}}{2}=3$. my drawing my idea From the equality we can ...
IONELA BUCIU's user avatar
1 vote
0 answers
55 views

How to prove triangle is equilateral without triangle congruence or similarity theorems?

I'm examining the following basic theorem: An isosceles triangle with vertex angle equal to $60^\circ$ has other two angles equal. (I know this is true for base angle as well, but I'm interested ...
Rusurano's user avatar
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2 votes
2 answers
171 views

How to find an acute angle of a right triangle inscribed in a square?

Working on Daniel J. Velleman. (2017). "Calculus: A Rigorous First Course" (p. 66) My question is focused on the purple circle on the image above. The solution given by the author is $\...
F. Zer's user avatar
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0 votes
2 answers
67 views

Does this theorem about two lines and two transversals have a name?

I've got this theorem right here. It involves two arbitrary lines $AB$ and $CD$ and two transversals $AC$ and $BD$, intersecting in between these lines. The theorem (or lemma, I don't know) in ...
Rusurano's user avatar
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1 vote
1 answer
28 views

Figuring out if an object is headed in the right direction at a crossing?

For a video game, I am trying to figure out whether the ship is headed to the right direction, that is, forward relative to the race direction as it's a racing game in 3 dimensions. Basically, the ...
aybe's user avatar
  • 307
0 votes
1 answer
36 views

Determination of the basis angle in a tilted equilateral triangle [closed]

I'm trying to solve a geometry problem involving intersecting lines and angles in an equilateral triangle. Could you please explain how to find the measure of the angle marked with a question mark, ...
Elif's user avatar
  • 21
2 votes
1 answer
33 views

Semantics of the angle between velocity vector and the positive $x$-axis

Let's say a particle moves in plane with curvature equal to $\kappa(t) = 2t$, with constant speed of $\|v(t)\| = 5$, such that $v(0) = 5\textbf{i}$, and the particle never goes to the left of the $y$-...
S11n's user avatar
  • 908
0 votes
2 answers
71 views

Bisector Intersection Proof [closed]

$AGF$ is a triangle, $B$ is the center of its inscribed circle, and $D$ and $E$ are the intersections of that circle with $[AG]$ & $[AF]$ respectively. I couldn’t figure out how to prove that the ...
Felix Shainker's user avatar
3 votes
3 answers
173 views

Sum of angles in a $1$-by-$3$ rectangle [closed]

This problem was in a competition for a job. It seems simple BUT the challenge is you cannot use trigonometry. Let there be 3 squares with side length of $\ell$ arranged in such a way that it forms a ...
ShrekLover's user avatar
-3 votes
1 answer
52 views

Circle Angle chasing problem: Compute APC+BQD [closed]

I was wondering if someone could give me a hint as to how to solve this problem: Let circle C1 intersect circle C2 at P and Q. A line intersects C1 at A and B, and C2 at C and D, such that the four ...
PabloGamerX's user avatar

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