# 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).

40 questions
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### 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.
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### 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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### “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 ...
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### 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? ...
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### 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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### 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 ...
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### 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 ...
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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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### 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 ...
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### 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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### 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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### 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 $$z = {2^4e^{i{\frac{4\pi}{3}}+2k\pi}}$$ ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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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### 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 $\alpha$ ...
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### 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 ...
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### 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 ...