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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).

4
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5answers
181 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.
9
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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.
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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 ...
4
votes
3answers
663 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? ...
12
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6answers
922 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?
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4answers
5k views

Do Zero degree angles exist?

If an angle is the measure of distance between to points (Edit: Ok, admittedly bad phrasing. A measure of rotation between two intersecting lines, or points, etc.), is there such a thing as a zero ...
1
vote
1answer
2k views

2D parametric equation for an arc between two points with a start angle

What's a parametric equation (eg. $(x,y)=(\cos(t \cdot 2\pi),\sin(t \cdot 2\pi)$ plots a circle where $t$ is the 'time' along the circle) that draws an arc between the two points $(x_0,y_0)$ and $(x_1,...
3
votes
6answers
15k views

calculating angle in circle

How to calculate angle in a circle. Please see the diagram to get the idea what I want to calculate? I have origin of circle that is $(x_1,x_2)$. I have a point on circumstance of circle that is $(x_2,...
1
vote
2answers
562 views

Interpolating Between 2 Angles

I'm trying to understand how this works, and mathematically I'm having difficulty. Given 2 angles between $(-2\pi, 2\pi)$: $\theta$ and $\phi$ I want to interpolate between them by the ratio: r. My ...
0
votes
4answers
50 views

Vectors and Norms

If $a$ and $b$ are vectors such that $\|a\| = 4$, $\|{b}\| = 5$, and $\|{a} + {b}\| = 7$, then find $\|2 {a} - 3 {b}\|$. I couldn't figure out how to start off this problem. I attempted to use $$\cos ...
16
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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 ...
9
votes
1answer
41k 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 ...
25
votes
10answers
5k 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.
9
votes
3answers
317 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 ...
3
votes
1answer
9k views

Quaternion - Angle computation using accelerometer and gyroscope

I have been using a 6dof LSM6DS0 IMU unit (with accelerometer and gyroscope) and am trying to calculate the angle of rotation around all three axes. I have tried many methods but am not getting the ...
2
votes
3answers
1k views

Find the two missing angles in a quadrilateral

This problem originates from a student who came asking for help. After spending some time, we couldn't solve this problem using (Euclidean) geometry alone. We had to resort to trigonometry to solve ...
1
vote
2answers
166 views

Angle trisection of $90^o$

Read on page #7 of article here, that angle of $90^o$ can be trisected. I went through this youtube video here, and here; and denote these two videos (denoting two separate methods) by (a), (b) ...
3
votes
1answer
10k views

Calculate 3D Vector out of two angles and vector length

What is the easiest way to calculate vector coordinates in 3D given 2 angles vector length? Input: Angle between X and Y axis: $$\alpha \in [0, 360).$$ Angle between Y and Z axis: $$\beta\in [0, 360)...
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votes
5answers
7k views

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 $dx=0$ ...
2
votes
3answers
46 views

Loops when drawing constantly changing angles and lines. [closed]

I start by drawing a line of 1 unit on the $x$ axis. I turn left (from the perspective of an ant on the line) by an angle of $\alpha$ and I draw a second segment of length $u$ from my endpoint of the ...
2
votes
4answers
206 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 $26.565^\circ$....
1
vote
1answer
352 views

Prove that the sum of angles is equal to 90° using complex numbers

On the picture, we see three squares: $ABGH$, $BCFG$ and $CDEF$. Prove that the sum of angles: $\angle DAE$, $\angle CAF$ and $\angle BAG$ is equal to $90°$. The real problem is that we have to ...
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vote
3answers
2k views

When do you take into account the +2kpi for complex numbers arguments in complex equations

$$z^2 = ({2e^{i{\frac{\pi}{3}}}})^8$$ To find z I took the square root of both sides which gives me: $$z = ({2e^{i{\frac{\pi}{3}}}})^4$$ which I rewrote as $$z = {2^4e^{i{\frac{4\pi}{3}}+2k\pi}}$$ ...
0
votes
0answers
28 views

Need help on a problem on trigonometry

In triangle ABC, AB=10, CA=12. The bisector of ∠𝐁 intersects CA at E, and the bisector of ∠𝐂 intersects AB at D. AM and AN are the perpendiculars to CD and BE respectively. If MN=4, then find BC. ...
2
votes
3answers
104 views

closed form expression for $\sin 10^o$? [closed]

As we know that $\sin 15^o$, $\sin 30^o$,$\sin 45^o$ have simple closed form expressions as these are multiples of 3, but i have never seen any simple closed form expression for $\sin 10^o$ or simply ...
1
vote
1answer
43 views

Height of a Triangle, and a Progression of Triangles

I am researching some algorithms and it turns out that the following figure I made can model what is happening in a "step". I am not a mathematician, so I was having a hard time with this one. The ...
1
vote
1answer
104 views

Why subtract $\pi$ in the definition of atan2?

Looking here the definition of the atan2 function is as follows: $$\operatorname {atan2} (y,x)={ \begin{cases} \arctan(\frac {y}{x}) & \text{if }x>0,\\ \...
1
vote
2answers
1k views

Length of two sides in a quadrilateral with given angles

I'm stuck finding the length of two sides in a quadrilateral for which I know all angles and the length of two sides. All red objects are know ($a,b,\alpha,\beta,\gamma $ and $\delta$). I need to ...
1
vote
2answers
214 views

Finding side-length proof in double-angle triangle.

In triangle $ABC$, $|AC| = b$ and $|AB| = c$. Angle $A$ is twice angle $B$. Prove that $$|BC| = \sqrt{b\cdot \left(b+c\right)}$$ I understand how to apply laws such as the cosine and sine law to this ...
1
vote
3answers
71 views

Find angle given ratio between sine and angle

I know an angle is between 0 and $\pi$ (180 degrees). I know the ratio between its sine and the angle itself. Specifically it's $\frac{15}{16}$, but I am more interested in the general case. Since ...
1
vote
3answers
2k views

Composition of Rotation and Translation in the Complex Plane — Finding Angle of Rotation and Point

A rotation about the point $1-4i$ is $-30$ degrees followed by a translation by the vector $5+i$. The result is a rotation about a point by some angle. Find them. Using the formula for a rotation in ...
1
vote
1answer
145 views

Geometry problem on angle bisectors and intersecting line segments

Two equal line segments $AB$ and $CD$ intersect each other at a point $M$. If the perpendicular bisectors of $AD$ and $BC$ intersect each other at the point $N$, prove that the two angles $\angle AMN$ ...
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vote
2answers
435 views

Proving the Secant Angles in the Circle

Ok, I know this is a very easy circle geometry problem, but I want to know that how to prove the theorem of angles in the circle. Like this image here: How can I prove that the angle $X$ is the half ...
0
votes
1answer
41 views

Bidirectionally of the “Tangent Criterion”

I've recently been reviewing some basic geometry concepts when I saw this one in Evan Chen's fantastic "Euclidean Geometry in Mathematical Olympiads" (EGMO). Proving $(i)\Rightarrow (iii)$ is quite ...
0
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2answers
59 views

Formal Definition for Angle Function

Which is the correct formal definition for the angle function $\theta(x,y)$ for a vector $(x,y)$ or complex number $x+iy$ with unit magnitude, such as $e^{i \theta}=x+iy$? Inverse trigonometric ...
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votes
1answer
67 views

How to calculate $B(x_1,y_1)$ when $\alpha$ and $A(x_0,y_0)$ are known?

In case one want to calculate $\alpha$ angle between $AB$ vector and $x$ axis, it can calculate $\alpha=arctan2(x_1-x_0,y_1-y_0)$. Thus, for example for $A(0,0)$ and $B(10,10)$ one will get $\alpha$ ...
0
votes
1answer
112 views

What are the effects of twists on a rectangular proportions?

Assuming we have a perfectly rigid rectangular cuboid object that cannot be stretched nor deformed in any way, but can be instantly reformed into a new arrangement so long as a specific set of ...
0
votes
1answer
99 views

Find $\angle{FPC}$: impossible question?

I was given this question by one of my friends: Shape $ABCD$ is a parallelogram. $AE = EB$ $BF = FC$ Find $\angle{FPC}$. but when I tried to solve it, I found it impossible without at ...
0
votes
3answers
871 views

How do you find the area of the shaded (gray) region of the square not getting overlapped by the circle or triangle [closed]

How do you find the area of the gray region in the problem. Pretty much the isosceles triangle is 2" tall and 2" wide at the bottom. The circle has a radius of 1" The square is 2" tall and 2" wide. ...
0
votes
1answer
159 views

Trisecting angle equivalence of constructing a segment

After reading Wikipedia and some previous questions asked in this site, I still don't understand this. Following the Pierre Wantzel. Triple angle formula cos(3theta ) and getting a polynomial p(x). ...