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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1answer
63 views

Prove this formula for the $\sin\left(\frac{x}{2^n}\right), x \in [0,\frac{\pi}{2}[, n \in \Bbb{N}$

The formula in question: $$\sin\left(\frac{x}{2^n}\right) = \sqrt{a_1-\sqrt{a_2+\sqrt{a_3+\sqrt{a_4+\dots+\sqrt{a_{n-1}+\sqrt{\frac{a_{n-1}}{2}\left(1-\sin^2(x)\right)}}}}}}$$ where $$a_k = \frac{1}{...
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1answer
93 views

What is it called when you subtract multiples of 360 degrees?

When you divide a vector by its magnitude to get a unit vector, there's a verb for that: you are "normalizing" the vector. Similarly, is there a verb for when you subtract multiples of $360$ ...
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1answer
44 views

Degree choice in improper integrals resulting in trigonometric functions

people. I have a question regarding the following improper integral, and others like it: $$\int_{-\infty}^\infty \frac{dx}{1+x^2}$$ The end result of that are the two limits: $$\lim_{a\to -\infty} \...
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2answers
177 views

What is the angle between the median and the bisector?

In a triangle ABC, what is the angle $\theta$ between the median AM and the bisector AD? I want a way to know the measure of that angle, given the lenghts of the sides and angles of the triangle. I ...
3
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1answer
514 views

Proof of $\angle$ sum of polygon.

First, I know this question might have been asked by several times, see here, for an example. Before someone may want to mark it as dulplicate, I would like to calrify what I want to ask. Mainly, I ...
3
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1answer
55 views

Why doesn't the average angle made by a function with the $x-axis$ work as expected?

The average angle made by a curve $f(x)$ between $x=a$ and $x=b$ is: $$\alpha=\frac{\int_a^b\tan^{-1}{(f'(x))}dx}{b-a}$$ I don't think there should be any questions on that. Since $f'(x)$ is the value ...
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2answers
4k views

Similar Triangles Problem (No corresponding sides)

This problem is from a Year 9 Oxford Maths textbook. I have tried to solve it since yesterday to no avail. Here are the questions - It is only (b/c) that I cannot solve, but I included the other ...
3
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1answer
372 views

Counter-clockwise angle between edges

I have a couple of connected lines (or rather edges) in the form of coordinates, that is for each edge a starting point $(x_s,y_s)$ and end point $(x_e,y_e)$. and I want to know a specific angle ...
3
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1answer
26 views

Angle of circular droplet

I am trying to find the angle $\theta$ of the following droplet: I think using $\tan$ is the right way to go, and I thought of using it on the angle formed by the line $r$ and $b$. However, that ...
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2answers
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How to calculate the quaternion from/and axis angle having parent and target position (camera and its target)?

I want to calculate the orientation (quaternion) of the virtual 3d camera that is looking at some point in 3d space. The illustration: According to this explanation the quaternion be calculated from ...
3
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1answer
859 views

Angle of tangent line and line $y=0,z=x$ is constant

Show that the tangent lines to the regular parameterized curve $\alpha(t)=(3t,2t^2,2t^3)$ make a constant angle with the line $y=0,z=x$. 1) The tangent line at each point is given, I believe, by $\...
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0answers
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Product of sines to sum

I stumbled across the following identity in a system I'm considering: \begin{align} \prod_{j=1}^N \sin a_j \end{align} which I need to rewrite as a sum (or otherwise) of sines or other expressions. $...
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2answers
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Stuck with a possibly impossible trigonometry question

I need to find the length of the arc between Y1 and Z1 in the image below. If you can even get me to the value of Y, then that will work. I appreciate the drawing may be crude, but imagine that Y and ...
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54 views

How can we calculate the sum of sines or cosines where the angles are in geometric progression?

For example: $$\cos\frac{\pi}{7} + \cos\frac{3\pi}{7} + \cos\frac{9\pi}{7}$$ In this example, there are only a few terms, and we can use things like $\cos(9\pi/7) = -\cos(2\pi/7)$ and complex ...
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142 views

How to find optimal solar panel angle using vectors?

Basically I am trying to find the optimum angle at which a solar panel should be installed by using vectors. I have done some research but found it a bit confusing , so basically I haven't got very ...
3
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0answers
53 views

Please explain how the following derivative graphically makes sense.

I have two vectors $\vec{A}$ and $\vec{B}$ as shown below: The point at the origin of vector $\vec{B}$ has coordinates $(x,y)$. The angle between the two vectors is $\theta$. Now in my physics book ...
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Formula for minimum distance for circles what won't touch each other depending on n?

Let's say that I have a circle, which is the "middle circle" (red in the pictures below). I also have a number (n) of identical circles, that should appear around the middle one, without touching. For ...
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1answer
195 views

Exact Dihedral angle for Disdyakis Triacontahedron

I've tried calculating the exact dihedral angle of a Disdyakis Triacontahedron, with no success. I cannot seem to find it online either. What is the correct approach to trying to figure out this value?...
3
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1answer
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Proving angles in the same corner equal

Suppose we have two line segments, AB and CD, which cross at point X. Now suppose there is an arbitrary point Y somewhere on the segment AX (that is, points A, Y and X are collinear). What is the ...
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4answers
84 views

Angles between vectors of center of two incircles

I have two two incircle between rectangle and two quadrilateral circlein. It's possible to determine exact value of $\phi,$ angles between vectors of center of two circles.
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2answers
331 views

Pythagoras: Get b when only a and angle α are given

Given the Pythagoras Theorem: a² + b² = c² Is there a way to get the value of b when we only have a value for a and the angle α? To be frank, I have no clue about that, what I want isn't the angle ...
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4answers
788 views

Why do we use degrees? [closed]

I see a lot of people who ask why we use radians instead of degrees. But why do we use degrees instead of radians. In the cases we use degrees instead of radians, what convenience does it bring? The ...
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3answers
106 views

Find the $\angle ACB$ of $\triangle ABC$.

If $PC=2BP$, $\angle ABC= 45^\circ$, and $\angle APC=60^\circ$, find $\angle ACB$. All solutions are acceptable but please try solving using reflection of point $C$ through the line segment $AP$. I ...
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4answers
263 views

How to convert components into an angle directly (for vectors)?

Let us say we have a vector with $x$-component $-2$, and $y$-component $-1$. We have the equation: $$\tan\theta=\frac{-1}{-2}$$ So if we take the inverse of $\tan$ of $\frac12$ we get $26.565^\circ$....
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5answers
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We have angle=arctan(dy/dx), but what happens when dx=0?

Here is a formula: $\text{angle}=\arctan(dy/dx)$. I can find an angle with my calculator for any value except $dx=0$. My question is: is there no angle or, is there something that says when $dx=0$ ...
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2answers
89 views

What is the value of $\angle x + \angle y$ in the following diagram?

$\angle p=30^\circ$, $\angle q=45^\circ$, $\angle r=50^\circ$, $\angle$ $s=25^\circ$. $\angle x + \angle y = ?$ Source: Bangladesh Math Olympiad 2016 junior Category I could not find any ways to ...
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3answers
198 views

How to evaluate $\cos{\frac{\pi}{8}}$?

I have to evaluate $\cos{\frac{\pi}{8}}$ and I'm supposed to do so evaluating first $\cos^2{\frac{\pi}{8}}$ (since it's an exercise to practice half-angle formulas). Solving this second formula I get ...
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2answers
89 views

Triangle Bisector Challenge

I've found this olympic geometry problem, which remains to be unsolvable by all my teachers and friends; In a triangle $ABC$ $[BE],[CD]$ are bisectors, $[BE]\cap[CD]=F$ $m(\widehat{FDE})=18^\circ$ $...
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2answers
68 views

Rotation Matrix and Triple Angle Formulas?

Define $R_{\theta}:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ as the rotation matrix by angle $\theta$, where $$R_{\theta} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \...
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2answers
118 views

Could we practically forgo education in degrees in favor of radians? [closed]

I've never been a fan of degrees and I'm still a bit resentful that my brain has been programmed to think in terms of them. Would it be practical to not teach them to children in primary education and ...
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2answers
71 views

Is there a way of determinine the side lengths of a isosceles triangle knowing its angles and area?

I want to be able to determine the side lengths (or at least one side length) of an isosceles triangle knowing only its surface area and angles. Is this possible?
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2answers
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Angles Formed By The Hands Of A Clock

Given a clock with hour, minute, and second hands that each move continuously (i.e., no “ticking” occurs), show whether there exists a time at which the lesser angle formed by each pair of the hands ...
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2answers
34 views

Interior angles of a polygon in the hyperbolic plane

This is fairly simple question about how to make sense of "angle" in the hyperbolic plane. The hyperbolic plane can be tesselated with pentagons, four to each vertex. In this tesselation, each ...
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2answers
99 views

How to use the Law of Sines to Find an Angle

I am trying to figure out how to find an angle with the law of sines. I have a triangle where: A = $120^\circ$ B = unmarked C = $\theta$ a = 45 b = unmarked c = 36 How can I find the angle ...
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2answers
283 views

How to find angle of arc given arc length and sagitta?

Given the length of an arc and the length of sagitta, can you calculate the angle (radians)? I struggle to work out all the parameters I need. For instance, to calculate the radius I need the length ...
2
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2answers
134 views

What is the word for the relationship between two angles that add up to 360˚

If A+B = 180˚ they are supplementary to each other but what would you call their relationship if they add up to 360˚?
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2answers
78 views

If a disk contains the diagonal of a quadrilateral, it will contain at least one of the vertices not lying on this diagonal

Final Version: Given a convex quadrilateral, then at least one of the two diagonals satisfy this: if a disk contains it, then it should contain one of the rest two vertices of the quadrilateral. 2nd ...
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1answer
122 views

Why isn't there an “hour” when measuring angles

So I was having a discussion about angles with a student today and they were given a problem like: convert 3$^{\circ}$ 15' 24" into degrees which is straightforward enough, $3 + \frac{15}{60} + \...
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3answers
51 views

can radius and degree exist in the same angle

I and my teacher had an argument about the result of $\sin(\pi-1)$. He said that to convert the angle from radius to degree you must replace every $\pi$ with 180 so he said that $\sin(\pi-1)=\sin(180-...
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1answer
36 views

Prove that the ratio of acute angles in a $3:4:5$ triangle is irrational

Inspired by a comment by @QC_QAOA on Question 3458920, which mentioned the ratio between the acute angles in a $3:4:5$ triangle, I would like to know if we can prove that this ratio is irrational. ...
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1answer
80 views

How to find the angle of reflection given 2 points and a mirror?

I'm pretty new to the Mathematics section of StackExchange and need some guidance on some math for a 2d topdown game I am making. In my game there is an Archer boss, who can shoot reflective arrows, ...
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1answer
213 views

If I square a value units of radians, is the result in units of radians squared or is it still radians?

I am writing a paper on circular motion. A function given is $$T=Msω^2L$$ The units for $ω$ are $\text{rad}/s$. What are the units for $ω^2$? Are they $\text{rad}^2/s^2$ or $\text{rad}/s^2$? If they ...
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4answers
285 views

Does the definition of the angle between two vectors require that they have the same “origin”?

I am thinking specifically about $\mathbb{R}^2$ so I can visualize things. By "origin" I mean that they start at the same point. When we graphically represnt vectors we don't care where the starting ...
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3answers
94 views

How to get the angle of the figure? [duplicate]

I have this problem, that can be solved with elemental knowledge. In order to challenge, I can't draw extra segments to solved it. This is the problem and i need to get the measure of $\angle{x}$ ...
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2answers
45 views

Find a distance based on height and angles towards base and top of an object?

Suppose I know the vertical height of an object and the angles (in relation to horizontal) towards the top and bottom of it. Is it possible to calculate the horizontal distance to the object based on ...
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2answers
234 views

What is angle subtended by two consecutive points on the circumference at the centre?

A circle is continuous and yet when you take two consecutive points it seems as if the angle subtended at the centre is zero. If there was some angle between them how could they be consecutive? Tell ...
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2answers
63 views

Find the angle NMC

In triangle $ABC$, $\measuredangle B = 70^{\circ}$, $\measuredangle C = 50^{\circ} $. On $AB$ and $AC$ take points $M$ and $N$ such that $\measuredangle MCB = 40^{\circ}$, $\measuredangle NBC= 50^...
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2answers
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Prove that for every $n\ge3$ there exists a convex $n$-gon with exactly 3 acute angles

I'm really not sure where to start. Induction can really be used, and that seems like the only way to prove for all $n$.
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2answers
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Find time, when angle between the minute and hour hands are given

The question which I was trying to solve is: Find the time between four and five o' clock when the angle between the hour hand and the minute hand is $78^\circ$. My approach: At four o' clock, the ...
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3answers
55 views

Is my observation correct about geometric constructions?

I have observed that it is possible to construct angles which are multiples of 3 with a ruler and a compass (Angles are in degrees). For example, 135°, 45° etc. can be constructed but Angles like 100° ...