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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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Solve for x, [cosx] ≤ [sinx]. where [.] is greatest integer function. [closed]

I solved this by graph but I can't do it mathematically.
Yug Bhupesh's user avatar
-2 votes
0 answers
31 views

Trignometric identies proving LHS =RHS [closed]

Prove $\csc \mathrm{A} - \csc \mathrm{A} \cos^2 \mathrm{A} = \sin \mathrm{A} $.
Pahan Induwara Liyanage's user avatar
1 vote
0 answers
59 views

Has anyone else seen the series $ \sum_{n=1}^{\infty}\left( \frac{a}{\sinh^2(an)}+\frac{b}{\sinh^2(bn)}\right)$, supposedly due to Ramanujan?

A while ago I saw a YouTube video that mentioned an infinite series and claimed that its closed form was due to Ramanujan. The specific series is: $$ \sum_{n=1}^{\infty} \left( \frac{a}{\sinh^2 (an)} +...
gabrielmfern's user avatar
1 vote
0 answers
30 views

What is the limit of this composed trig function (first question)? [duplicate]

This is my first question here so I don't know if I formatted this well. Please let me know how I could improve. So, I was messing around on desmos, specifically with composing functions n times. An ...
Sebas31415's user avatar
-3 votes
0 answers
80 views

Find $\arcsin c$, $\, c\in\Bbb C$ [duplicate]

A math fan sent me a solution of the weird equation $\sin z=2$ posted in Quora. It is Weird because in real calculus, we experienced that $-1\leq \sin x\leq 1$. I saw this question in so many places ...
Bob Dobbs's user avatar
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0 answers
25 views

Trigonometry series question involving sine θ [closed]

sin θ - sin 2θ + sin 3θ - sin 4θ +....... n terms Write in simplified form
Ridham's user avatar
  • 1
-1 votes
1 answer
22 views

$f(x) = \tan(1/x)$, for $f(x) = 0, x$ must equal $1/nπ$ where $n$ is any integer. [closed]

Why must $n$ be positive in part a and nonnegative in part b? I see $n$ can be any Integer at both parts. My problem is that my calculus book says $n$ must be positive, but I don't see it this way. ...
Ahmed Adel's user avatar
0 votes
0 answers
43 views

Find value of $\theta$ from an expression of $\sin\theta$ involving several terms

My expression for $\sin\theta$ reads $$\sin\theta = bx + \frac{2ac}{b^2} $$. And the resulting expression for $\theta$ is $$\theta = \sin^{-1}(bx) + \frac{2ac}{b^2}\frac{1}{\sqrt{1-b^2x^2}} $$. Now it ...
Argentum's user avatar
0 votes
0 answers
32 views

acute angles $\alpha$ and $\beta$ of the triangle $ABC$ satisfy $\sin^2 \alpha + \sin^2 \beta = \sin (\alpha + \beta)$, then $ABC$ is right-angled. [duplicate]

Given that the acute angles $\alpha$ and $\beta$ of the triangle $ABC$ satisfy the condition $\sin^2 \alpha + \sin^2 \beta = \sin (\alpha + \beta)$. Prove that the triangle $ABC$ is right-angled. ...
Mark A.'s user avatar
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Reference for the lemma that, given $0<A<180^\circ$, function $f(x)=\frac{\sin x}{\sin(A-x)}$ is injective for $0\leq x\leq A$

There is a geometry lemma used in my mathematical community which states: Given an angle $A$, $0 < A < 180^\circ$, the function $$f(x) = \dfrac{\sin(x)}{\sin(A-x)}$$ is injective, for $0 \leq x ...
TheSega's user avatar
  • 27
0 votes
1 answer
51 views

Is it the case that $\cos(\arccos(z))=z \iff z\in[-1,1]$? Is my derivation of the falsity of that statement incorrect?

I was recently told that $\cos(\arccos(z))=z \iff z\in[-1,1]$ $\tag1$ I came to a different conclusion by using the following reasoning: Let $z\in\mathbb{C}$ Let $\operatorname{Log}(z)$ denote the ...
Simon M's user avatar
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0 answers
80 views

Determine all the values of $a^2$ such that the following equation has exactly one solution in $t$

Let $ p(t) = A + B \cos t + C \sin t $ where $A,B,C \in \mathbb{R}^3 $, be a given ellipse. Now consider the following equation $ p^T (a^2 I - u u^T) p = a^2 (a^2 - c^2) \tag{1}$ where $u$ is a known ...
Quadrics's user avatar
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-3 votes
0 answers
69 views

(Solved!) Hard 3-Dimensional Trigonometry Problem [closed]

I have had difficulties solving part (e) of this question: in part (a) I used the fact that angle AVC was a right angle, then made a perpendicular dropping from V to the center of the square base (I ...
Tanish Shukla's user avatar
1 vote
2 answers
104 views

Find the angle $ECA$

I found this puzzle some time ago on the Internet. Triangle $ABC$ is like on the picture: where $AD$ is an altitude. One need to find angle $ECA$. How to solve it? Insane attempt: $\left|\angle EBD\...
Piotr Wasilewicz's user avatar
-1 votes
0 answers
19 views

find the intersection point with the cube on the ray [closed]

find the intersection point of the ray with the cube, having the ray direction
Daniil Mikhailichenko's user avatar
2 votes
1 answer
58 views

Ratio of parts of triangle-side divided by line connecting vertex to circumcenter

Say we have a circumscribed triangle $ABC$ with circumcenter $O$. Say also that $D$ is the intersection of the (infinite) line $AO$ with $BC$. I've been trying to determine the ratio of $BD$ and $CD$ ...
AnonA's user avatar
  • 87
-1 votes
0 answers
39 views

How to calculate volume of the cylinder [closed]

Recently in class we were 3d modeling a monument and this problem came up: given a cylinder intersecting with a pyramid at the base, the angle alpha base to the height being 90° and both other angles ...
kian's user avatar
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0 answers
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Finding the equation of a sine wave using two points and a tangent [closed]

I am trying to find a sine wave of form y=asin(b(x-c)) (a=/=0, b=/=0) that intersects the point (45, -25) with a slope of 1.55, and intersects the x-axis at x=52. using this information I made 3 ...
mmmm's user avatar
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20 votes
1 answer
1k views

Integrating a large product of sines

Recently, I came across the following integral: $$\int_{0}^{2\pi}\sin(x)\sin(2x)\sin(3x)\sin(4x)~\mathrm dx=\frac{\pi}{4}$$ which can be easily solved by some trigonometry. But when trying to find a ...
pvr95's user avatar
  • 529
2 votes
2 answers
97 views

Prove that the equation $\sin(x) \cdot \sin(2x) \cdot \sin(3x) = \frac{3}{4}$ has no solutions. [closed]

Prove that the equation $\sin(x) \cdot \sin(2x) \cdot \sin(3x) = \frac{3}{4}$ has no solutions in real numbers. Attempt: I will know how to solve such an equation if there was a number 1 on the right ...
user avatar
2 votes
2 answers
86 views

$\sin$ of sum of two angles proof error

I'm doing a refresher on high school math and wanted to prove the $\sin(\theta + \phi) = \sin(\theta)\cos(\phi)+\cos(\theta)\sin(\phi)$ identity but I can't get it right. I drew two right triangles $...
AJB's user avatar
  • 121
0 votes
3 answers
54 views

How to use Givens rotation for complex matrix?

I have a $A$ an hermitian matrix and i want to tridiagonalize it with givens rotation. I found this : QR factorization of complex matrix I try it with this matrix : $ A = \begin{bmatrix} 4 & 2+...
superneiluj's user avatar
2 votes
3 answers
178 views

Calculating percentage coordinates on an arc

Please forgive me, I'm not a math geek, I'm a Linux guy that's trying to do something cool. So straight out the gate, I do not understand advanced (or even what you might deem quite simple math). I ...
Jim's user avatar
  • 123
-2 votes
1 answer
92 views

Couldn't Figure Out What is Wrong with My Solution [closed]

My Solution I am currently working on a proof and to express it in a clearer way I used Manim(a math animation software) but for some reason it did not work. So I have to wonder is there anything ...
Secret Mushroom XXX's user avatar
4 votes
1 answer
372 views

Maximizing with Cauchy-Schwarz inequality

I want maximize the function $f(x)=\cos(x)+\sin(x)\cdot\cos(x)$, with $x \in (0,\frac{\pi}{2})$. By derivation $f'(x)=0 \Rightarrow x=\frac{\pi}{6}$. But, if we write $f(x)=\cos(x)+\frac{1}{2}\sin(2x)$...
Cgomes's user avatar
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-1 votes
0 answers
40 views

Given that a right angle triangle has an hypotenuse of 10 cm and an opposite of 8 cm calculate the angle within [closed]

I put Taylor's of 10 cm and an opposite of 8 cm a right angle triangle
Oluwakemi Abiodun's user avatar
-2 votes
0 answers
47 views

nth derivative recursive formula

I'm trying to find a recursive series representation of the $n$-th derivative of the following function. $D^{(1)}_{a, b}(x) = b\sqrt{\frac{p}{q}}D_{a + p - 1, b - 1}(x) - a\sqrt{\frac{q}{p}}D_{a - 1, ...
Ghull's user avatar
  • 81
-1 votes
0 answers
36 views

The side of the cube that is on the vector (check the ray solution for each side, I do not want to run rays along the vector) [closed]

I have a cube, preferably having only its center, and a vector (which is just an infinite ray that has directions to rotate in space (normalized)) which looks somewhere in space, if there is a cube, ...
Daniil Mikhailichenko's user avatar
2 votes
1 answer
73 views

Solve the equation: $ \log_{\sin x} (\cos x) - 2 \log_{\cos x} (\sin x) + 1 = 0. $ [closed]

Solve the equation: $ \log_{\sin x} (\cos x) - 2 \log_{\cos x} (\sin x) + 1 = 0. $ Attempt: I transorm this equation in $(\log\cos x-\log\sin x)(\log\cos x+2\log\sin x)=0$, therefore $\cos x=\sin x$ ...
user avatar
0 votes
0 answers
37 views

Lissajous curve with irrational argument

Can a lissajous curve with irrational argument such as $$x=\cos(t) \\ y=\sin(t\sqrt{2})$$ be used as a $1$ to $2$ mapping since for any two given $x$ and $y$ in a rectangular region, $-1$ to $1$ and $-...
Ben North's user avatar
-3 votes
0 answers
52 views

Relationship between $\beta$ and $\alpha$ when $\tan (\beta) = n \cdot \tan (\alpha)$ [closed]

Given the equation $\tan (\beta) = n \cdot \tan (\alpha)$ is there perhaps a general relationship between $\beta$ and $\alpha$ that can be expressed as formulas or maybe e.g. a hyperbolic scale? The ...
Gerard's user avatar
  • 1,527
7 votes
3 answers
203 views

Solve the equation $9x\sqrt{1-4x^2} + 2x\sqrt{1-81x^2} = \sqrt{1-144x^2}$

I need to solve the equation $$9x\sqrt{1-4x^2}+2x\sqrt{1-81x^2}=\sqrt{1-144x^2}$$ over real numbers. Squaring both sides and transforming this in 4-degree equation I got $x = \frac{1}{16}$. But can ...
Ivan Borisyuk's user avatar
0 votes
2 answers
71 views

Smallest integer $n$ so that $\tan(5\pi/16) - 2\tan(\pi/8) = \tan(n\pi/16)$

So the question goes like this: If $\tan(5π/16) - 2\tan(π/8) = \tan(nπ/16)$, then smallest positive integer n is equal to ? Try 1: I tried simplifying the LHS in terms of $\pi/16$ by converting $\tan(\...
PXsmath's user avatar
0 votes
2 answers
35 views

Extreme Values of an Expression with Symmetry in Sines and Cosines [duplicate]

I've been doing a lot of trigonometry-based sums lately, and came across this one: I simplified this to obtain: $$\frac{1+\sin{x}+\cos{x}}{\sin{x}\cos{x}}$$ Now, on further solving, I realized that ...
Schrödinger's Cat's user avatar
0 votes
1 answer
23 views

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
1 answer
85 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
  • 107
1 vote
2 answers
72 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
  • 105
1 vote
1 answer
69 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
89 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
  • 73
4 votes
1 answer
176 views

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
1 vote
1 answer
61 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
227 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 ...
Quadrics's user avatar
  • 24.2k
0 votes
1 answer
81 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
0 votes
0 answers
55 views

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
95 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
272 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 ...
Circuit Sage's user avatar
1 vote
1 answer
51 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
78 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
81 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
  • 766
1 vote
0 answers
30 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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