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Questions tagged [trigonometry]

Questions about trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles and other topics relating to measuring triangles.

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When is function of bounded variation piecewise monotone on finite number of subintervals?

Given is a function $f:I \rightarrow \mathbb{R}$ on a closed and bounded interval $I \subset \mathbb{R}$. If $f$ is of bounded variation on $I$, that is $\text{Var}_I(f) < \infty$, then we know ...
Nelus127's user avatar
-1 votes
0 answers
43 views

Tricky logarithm question involving three equalities and different bases.

I have the following statement given to me: If $\log_{2}a=\log_{12}b=\log_{16}\left(a+b\right)$, and $\frac{b}{a}=2\sin\theta$, then where does $\theta$ DOES NOT lies: [a, b and $\theta$ all belongs ...
DevMayukh's user avatar
0 votes
1 answer
68 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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1 vote
2 answers
51 views

Show that $\int_{(0, 2\pi)} \frac{\cos(2\theta) d\theta}{5+3 \cos(\theta)} = \frac{\pi}{18}$ using residue theory

Let $\theta=e^{iz}$ with $C : |z| <= 1$ it implies $d \theta = \frac{dz}{iz}$ and then we can get $\cos \theta = \frac{1}{2} (z+\frac{1}{z})$ and $\cos(2\theta) = \frac{1}{2}(z^2+\frac{1}{z^2})$ so ...
Ocean's user avatar
  • 99
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0 answers
26 views

Understand the upper bound of $\sum_{j=1}^n\cos(\theta_1-\theta_j)$

Consider $\theta$, a $n$-dimensional vector $\in\mathbb R^n$, belongs to the region $$[0,2\pi]^n/\ [-\pi/2,\pi/2]^n$$ Thus, this region is the unit circle except for all $\theta_i$ are in the one ...
chloe's user avatar
  • 576
-7 votes
1 answer
64 views

Prove that $64\sin(π/42)\sin(5π/42) \sin(13π/42) \sin(17π/42) \sin(19π/42) \sin(31π/42) = 1$ [closed]

Should I proceed further with the formula for difference of cosines?
Aryan Arora's user avatar
1 vote
1 answer
60 views

Integral of $\tan(A+B)\tan(A-B)$

Evaluate $\int \tan(2x+a)\tan(2x-a) dx$ I actually got to this integral while trying to solve for the actual function $\frac{1}{\cos(2x+a)(\cos(2x-a))}$. I multiplied and divided by $\cos 2a=\cos((2x+...
a_i_r's user avatar
  • 681
0 votes
2 answers
85 views

How to find angle for area of triangle

I was able to figure out $|OP| = \frac{1}{\sqrt{2}}$ and that $P\hat{Q}O = \frac{\pi}{3}$ and it seem to be intuitive to me that $O\hat{P}Q = \frac{\pi}{2}$ but I'm not sure what's the "best&...
Ally's user avatar
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0 answers
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Determining the angle of a point in an image taken by a camera

Assume I have a certain picture taken aerially by a drone. The yaw, pitch, and roll of the drone gimbal are known, and so are the horizontal and vertical FoV of the camera. We also know the $x$, $y$, ...
Samuele B.'s user avatar
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0 answers
45 views

Cosine as nested roots

I have been playing around with circles lately, and I have found an interesting limited relationship between prime factors and cosine. Have the form of: $$\cos{\left(2\pi\frac{1}{p}\right)}$$ And that ...
John Clement Husain's user avatar
3 votes
6 answers
176 views

Rotating and scaling an arbitrary triangle such that the new triangle has its vertices on the sides of the original one

Given $\triangle ABC$, and a scale factor $r \lt 1 $, I want to find the necessary rotation (center and angle) such that the rotated/scaled version of the triangle has its vertices lying on the sides ...
c'est pas normale's user avatar
-6 votes
0 answers
63 views

Drawing the graph of $\frac{\sin\{x\}}{\{x\}}$, where $\{\}$ is the fractional part function [closed]

How to draw the graph of $\dfrac{\sin\{x\}}{\{x\}}$, where $\{\}$ is the fractional part function?
MEGHAVI THAKKAR's user avatar
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24 views

Non-fitting scaling and rotation of polygons

Is there a scaling factor $r < 1$ and a rotation angle $\psi \in [-\pi, \pi]$ such that no triangle exists that, after being scaled by $r$ and rotated by $\psi$, can fit into its original shape? If ...
trurl's user avatar
  • 11
0 votes
1 answer
31 views

Finding the smallest number of sides for a regular polygon in which two diagonals form an angle of $50^\circ$ [closed]

I need some help on this problem: In a regular polygon there are two diagonals such that the angle between them is $50^\circ$. What is the smallest number of sides of the polygon for which this is ...
mathisdagoat's user avatar
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1 answer
72 views

How to prove that $\sin\left(\frac{\pi}{2n}\right)\sum_{k=1}^{n}\sin\left(\frac{2k-1}{2n}\pi\right)=1$? [closed]

How to prove that $$\sin\left(\frac{\pi}{2n}\right)\sum_{k=1}^{n}\sin\left(\frac{2k-1}{2n}\pi\right)=1$$ ?
El Mismo Sol's user avatar
-2 votes
0 answers
58 views

Proving $\sin(x) + \sin(y) + \sin(z) = 4 \cos(x/2) \cos(y/2) \cos(z/2)$, for $x$, $y$, $z$ the angles of a triangle [closed]

My fellow classmates and I have been struggling to prove a trigonometric identity, and we were wondering if someone on this forum was willing to help. Prove that for a triangle with angles $x, y, z: \;...
Chips's user avatar
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0 answers
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If $N$ is odd, is $\sum_{k=1}^{(N-1)/2}\frac{1}{\sin^2(\pi k/N)}$ always rational? [duplicate]

For an odd number of $N$, derive the following $$\sum_{k=1}^{(N-1)/2}\frac{1}{\sin^2(\pi k/N)}$$ Numerically, it seems to be a rational number, but I can't prove it. For $N=3$, the answer is $4/3$. ...
Yoshiki S's user avatar
3 votes
2 answers
93 views

Is this provable? $\lim_{x\to 0} \frac{\sin (\pi \cos^2 x)}{x^2}=\pi $ [duplicate]

I came across this question $$\lim_{x\to 0} \frac{\sin (\pi \cos^2 x)}{x^2}=\pi $$ I tried following method simplify it into a $x\to0$, $\sin x / x$ type limit $$\lim_{x\to 0}\frac{\sin(\pi \cos^2 x)}{...
donthababakka's user avatar
3 votes
2 answers
261 views

A peculiar problem on geometry relating to finding the angle between the diagonals of a cyclic quadrilateral

A quadrilateral with side lengths $a$,$b$,$c$ and $d$ can be inscribed in a circle such that $a=\frac{1}{c}$ and $b=\frac{1}{d}$. If $∆A$ represents the area of the quadrilateral. Prove that the angle ...
MIND FORGE NEXUS's user avatar
1 vote
0 answers
26 views

Why is this set of functions compact in L^1 (in proof of localization principle by Katznelson)

These are the hypotheses of the localization principle, a theorem in Katznelson's Introduction to Harmonic Analysis (to be found in chapt.2, section 2): Let $f$ be a complex-valued periodic function ...
Ulysse Keller's user avatar
-1 votes
1 answer
77 views

What's wrong in this proof- $\frac{\tan(3A)}{\tan(A)}$ coming out to be $1$

I was looking to simplify the expression $$k = \frac{\tan(3A)}{\tan(A)}$$ I proceeded as follows: $$\begin{align} \tan(3A) &= \frac{\sin(3A)}{\cos(3A)} \\[4pt] \sin(3A) &= 3\sin(A) - 4\sin^3(A)...
Smarika Singh's user avatar
3 votes
5 answers
75 views

Minimizing $\left(\frac{c}{a} + \frac{c}{b}\right)^2$, where $c$ is the hypotenuse of a right triangle with legs $a$ and $b$

This question is regarding the following problem Given that $a, b, c$ are the sides of the $\triangle ABC$ which is right angled at $C$, then what is the minimum value of the following expression? $$\...
koiboi's user avatar
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1 vote
0 answers
29 views

Given a sine and cosine function that define a circle with radius 1, is there a way to generate a corresponding distance function?

I'm aware that there is a branch of math about this, but I don't quite remember the name. I was thinking about this because I was messing with the functions $\sum_{n=1}^{\infty}(\frac{\sin(xn^2+t)}{n^...
PythonBoi's user avatar
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48 views

Trigonometry book that complements "A Course of Pure Mathematics" by G.H. Hardy.

While there are countless threads suggesting books on Trigonometry, there are none that complement the style of Hardy's book. What I mean is: a) The style being more illustrative of what is going on ...
noobman's user avatar
  • 155
0 votes
1 answer
43 views

Is there transformed unit circles that represent transformed trigonometric functions?

I was doing some questions where we find general solutions for transformed trigonometric functions and I became deeply focused on the transformed general equation for such transformed functions. This ...
Rohan Rajasekar's user avatar
-2 votes
5 answers
110 views

How to solve $5\sin x+2\cos x=5?$ [closed]

How to solve this types of equation? If I square both side equation will change. How to solve with trigonometric formulas?
Arman Abbw's user avatar
1 vote
2 answers
97 views

Applying Derivative Formula for $\frac{d}{dx}\frac{1}{\tan x}$

Using the identity, $\cot x = \frac{\cos x}{\sin x}$, I used the derivative formula for limits. This is done as followed: \begin{align} \bigg(\frac{\cos x}{\sin x}\bigg)^{'} &= \lim_{h \to 0} \...
BeaconiteGuy's user avatar
4 votes
2 answers
87 views

Using Sinus or Cosinus theorem gives an other result. What's wrong?

Suppose we are in the following situation : So $AB= 10$ cm and $BC=5$ cm, and $\beta =43$ degree. So using Cosinus theorem, I get $$AC= \sqrt{AB^2+BC^2-2AB\cdot BC\cdot \cos(\beta )}=7,20\ cm$$ Now ...
joshua's user avatar
  • 1,097
3 votes
1 answer
87 views

Calculate $ \lim_{n \to \infty} \left( \sum_{k=1}^n \left( \sqrt{n^4 + k} \cdot \sin \left( 2\pi \cdot \frac{k}{n} \right) \right) \right) \ $

$$ \mbox{What is the value of this limit ?:}\quad \lim_{n \to \infty}\sum_{k = 1}^{n}\sqrt{n^{4} + k\,}\ \sin\left(2\pi\,\frac{k}{n}\right) $$ I tried looking for sum Riemann sums first, nothing. I ...
Stefan Solomon's user avatar
3 votes
2 answers
615 views

Higher Degree Differential Equation

I'm having trouble with the development of this equation: $$\sin{y'}-x=0 $$ If I solve the equation with this method, I don't have any problems. $$\sin{y'}=x$$ $$y'=\arcsin{x}$$ $$\int dy=\int \arcsin{...
Bass's user avatar
  • 51
-1 votes
1 answer
131 views

Finding the number of solutions to $x=1964\sin x-189$

Find the number of solutions of the enigmatic equation $$x=1964\sin x-189$$ We define $$f(x)=\frac{x+189}{1964} \qquad\text{and}\qquad g(x)=\sin x$$ We can see that $f(x)$ is monotonic and $|g(x)|\...
MathStackexchangeIsMarvellous's user avatar
1 vote
2 answers
77 views

Can't find all solutions to a trigonometric equation

I have a problem to solve: Plot the function $2 \cos(6x) \sin(3x)$ and graphically find the maximum and minimum points. Find the maximum and minimum points analytically. Compare the results. My ...
x2t's user avatar
  • 21
4 votes
2 answers
63 views

Ratio of radii in rings of tangential circles [closed]

The picture of my table mat above is the inspiration for this problem. Suppose we have a central circle. Around this central circle we lay a ring of $n$ smaller circles of radius $r$ such that they ...
Cristof012's user avatar
3 votes
0 answers
54 views

All real solutions of trigonometric equation

Find all real number $a$ ($0\le a<2\pi$), such that there exists two real numbers $b,c$ satisfying the following three conditions: $a<b<c<2\pi$ $2b=a+c$ $2\cos(a)=\cos(b)+\cos(c)$ What I’...
coder114514's user avatar
1 vote
1 answer
67 views

$ \arctan\left(\frac{\sin\alpha + A\cos\alpha}{\cos\alpha - A\sin\alpha}\right) = \alpha + \arctan A $. Is this in fact an identity?

I found the identity at the bottom in a paper titled Nonlinear Estimation with Radar Observations by Miller and Leskiw: I tried to prove it but I'm at a loss. Much ...
Lukasz's user avatar
  • 19
0 votes
1 answer
24 views

Algorithm / equations to position a point just outside or inside the edge of a regular polygon?

Here is a polygon with a dot inside an edge, and a dot outside another edge. How do you calculate the $x$ and $y$ position of any dot (whether it's inside or outside of the line's edge) positioned ...
Lance's user avatar
  • 3,763
0 votes
0 answers
18 views

Converting from actual distance to pixel distance in zenith aerial drone image

I have a set of zenith pictures taken by a flying drone. There are some points of interest in the image for which I know the distance from the center of the image in meters. For each one of those ...
Samuele B.'s user avatar
0 votes
0 answers
36 views

Trigonometry question regarding circumcircles of triangles

Question is as follows: If O is the circumcentre of triangle ABC, prove that the ratio of the radii of the circumcircles of triangles ABC, OBC is $2 \cos A : 1$. Not sure how to get started on this ...
GR L's user avatar
  • 341
0 votes
0 answers
40 views

Checking 'No-resonance' condition for the eigenvalues of a discrete Laplacian matrix with Dirichlet boundary condition

1-D discrete Laplacian matrix (finite difference scheme) has eigenvalues as (page-2 in ref.): $$\lambda_j = sin^2(\frac{j\pi}{2(N+1)});\ j\in \{1, 2, ..., N\}$$ Where $N$ is the number of ...
Manish Kumar's user avatar
0 votes
1 answer
27 views

Equations to rotate and position triangles around a regular polygon?

What equations can one use to orient and layout the triangles on any regular pentagon from 3 to let's say 12 sides? Like this: Here, we have 3 at side 1, 1 at side 3, and 2 at side 5. The triangles ...
Lance's user avatar
  • 3,763
2 votes
0 answers
36 views

Equation of how to flip a polygon 180 degrees, and place nested inside parent polygon?

I am trying to position and scale this black pentagram inside the gray one perfectly, but haven't figured out the right equation. There are two parts to solving this: How do you determine the inner ...
Lance's user avatar
  • 3,763
1 vote
1 answer
45 views

Number of solutions of parametric equation $\sin\left(a(\sin x+\cos^2x)\right)=0$ for $a\in\mathbb{R}^+$ and $0\leq x\leq\frac{\pi}{2}$

I am trying to find the number of solutions $N(a)$ of the following parametric equation: $$\sin\left(a(\sin x+\cos^2x)\right)=0,$$ where $a\in\mathbb{R}^+$ and $0\leq x\leq\frac{\pi}{2}.$ What I have ...
lorenzo's user avatar
  • 4,114
1 vote
0 answers
84 views

Is $c^2+s^2=1$ the only algebraic identity in the trigonometric algebra? [duplicate]

Let $\mathbb{R}[\cos(x),\sin(x)]$ be the algebra generated by the cosine and sine functions over the real numbers. Is it true that it is isomorphic to $$\dfrac{\mathbb{R}[c,s]}{\langle c^2+s^2-1\...
Qfwfq's user avatar
  • 983
-2 votes
0 answers
26 views

Multiple angle formulas for tangent

There are direct multiple angle formulas for sine and cosine, in terms of Chebyshev polynomials. For example, the cosine of $n\theta$ is $$ \cos{n\theta}=T_n(\cos\theta) $$ Where $T_n$ is the n-th ...
Francesco Sollazzi's user avatar
1 vote
1 answer
68 views

How to see the graph of this? $n=\sin(x)\cos(y)+\sin(y)\cos(z)+\sin(z)\cos(x)$ [closed]

$$n \text{ (constant)} = \sin(x)\cos(y)+\sin(y)\cos(z)+\sin(z)\cos(x)$$ Code is this, but I can't emulate this. Why? Even WolframAlpha can't make graph too. ...
Myeongjun Chae's user avatar
0 votes
1 answer
33 views

In concentric circles Triangle formed from intersection of a line making 45 degree with x axis where inner circle meets x axis to outer circle [closed]

I have two concentric circles one of radius 5 cm and outer one of 10 cm, their centers being 0,0 I want to calculate P B and H of the triangle formed by intersection of a line on outer circle making ...
Ken Kaneki's user avatar
1 vote
0 answers
65 views

A solution to an equation on trigonometric functions

Consider the equation $$ 2 \sin 2x + \cot \frac{(n-1)x}{2} = 0 $$ on $x$ for fixed positive integer $n$. Is there an explicit solution or an approximate solution for $x$ in each interval $(\frac{2k\pi}...
user1150713's user avatar
3 votes
1 answer
103 views

Is $f(x) = \sin x$ the unique function satisfying all five: $f(0)=0;\ f'(0)=1;\ f(\pi/2)=1;\ f'(\pi/2)=0;\ -1\leq f''(x)\leq 0$ for $x\in [0,\pi/2] ?$

I would like to prove or find a counter-example to the following proposition (which I came up with), please. Suppose $f:[0,\pi/2]\to [0,1]$ is twice differentiable in the interval $[0,\pi/2]$. ...
Adam Rubinson's user avatar
0 votes
0 answers
102 views

Solving the equation $\sin x-\sin(2x)-\cos(3x)=0$

I need the solution for the equation $\sin x-\sin(2x)-\cos(3x)=0$. It has to be solved using trigonometric identities (not by graph). I have tried using the identity $\sin x-\sin y=2\sin\left(\frac{x-...
Xabcd's user avatar
  • 21
1 vote
1 answer
46 views

Solving a trigonometric equation originating from law of sines

Often when using trigonometry, especially the sine law to solve geometry problems, I end up with the following equation sin (x) sin(a) - sin (x+b) sin(c) = 0 However, I am often stuck as it is common ...
Jelly Qwerty's user avatar

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