Questions tagged [trigonometry]

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

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3 views

Find the perimeter of a four-sided polygon that is formed between two triangles

Problem Statement An acute triangle intersects with another triangle to form a four-sided polygon. Given all angles, and given the distance of two sides of the polygon, find the other two sides of ...
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2answers
29 views

Sine function and wave questions

I have been doing research for a few weeks, and I was trying to learn more about sine. I learned how it was used in trigonometry for calculating angles. It is the same as the sine function, which is ...
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27 views

Trigonometric substitution in an indefinite integral

I want to consider a trigonometric substitution to solve the following integral: $$\int \frac{1}{x\sqrt{1-x^2}}\, dx$$ The substitution is $x=\sin{t}\,dt$ and so $t=\arcsin{x}$ with $t\in[-\frac{\pi}{...
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Extraneous solutions without performing squaring operation

Solve in $[0, 2\pi]$ $$\sec x+\tan x=2\cos x$$ My mind boggled while solving the trigonometric equation in two different ways: Method $1.$ Assuming $x\ne \frac{\pi}{2},\frac{3\pi}{2}$ We have: $$\sec ...
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1answer
33 views

How to solve trig function

I have the following function $f(x)$ and known values $a$ $f\left(x\right)=\cos\left(\cos^{-1}\left(\frac{1}{2\sqrt{\frac{1}{2}-x+x^{2}}}\right)-\cos^{-1}\left(\frac{x}{\sqrt{\frac{1}{2}-x+x^{2}}}\...
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3answers
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Law of tangents with an angle bisector without knowing angles

"In $\triangle ABC$, let D be a point on BC such that AD bisects $\angle A$. If AD=6, BD=4, and DC=3, then find AB" This problem is from the Mu Alpha Theta 1991 contest, it appears in volume ...
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1answer
53 views

Two trigonometry homework problems - details appreciated!

I'd appreciate a solution to these two related problems with a figure, if possible. I understand that it may involve trigonometry, but I can't be completely sure. John is buying a 6 foot tall ...
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Strong approximate for $\sin\frac{{\pi}}{360}x $

If we have $s$ semicircle with the diameter $AB$ (with length $1$) and the center $O$, then we can approximate $\operatorname{chord} AC$ where $x$ represents the value of the $\angle AOC$, in degrees. ...
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How to express $\arcsin 2x$ in terms of $\tan $? [closed]

Someone please help me. How to express $\arcsin 2x$ in terms of $\tan $? Thanks.
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Should the probability of $f(x)>0$ be the same for any choices of $A,B$?

We have this function for $x>0$ and two constants $B>A>0$ $$f (x)=8 \cos (A x+B x)+19 \cos (A x-B x)+5 \cos (2 A x-B x)\\\quad+8 \cos (A x-2 B x)+2 \cos (2 A x-2 B x)+19 \cos (A x)\\+2 \cos (...
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Where does the R dθ triangle side length come from in this Coulomb's Law, infinite line charge example?

I'm reading Purcell & Morin's E&M 3rd edition. In section 1.12, they demonstrate how Gauss's Law can save effort over Coulomb's by solving the same "infinite line charge" example ...
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1answer
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Trig 101: Calculate coords of point P such that it is distance n from line AB and distance m from line BC

I've been in software engineering nearly 25 years and my trigonometry and maths skills are very rusty. We have a requirement to drill a hole in the corner of a sheet of material, where the corner isn'...
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34 views

How can I determine what objects an object can “see” in 2d space?

I am working on some software. I apologize for not putting this in any kind of real notation. I have a series of points (x,y) and facings (θ). How do I find if a given point + facing is looking at ...
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Can someone please give a hint on this complicated problem I have got stuck with it for a long time?

I have this function for $x>0$ and two constants $B>A>0$ $$f (x)= \left(2 \cos \frac{x (A-2 B)}{2} +\cos \frac{A x}{2}\right) \times\left(\cos \frac{x (A-B)}{2} +2 \cos \frac{x (A+B)}{2} \...
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5answers
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Finding sum of the roots of $(\sin x+\cos x)^{(1+\sin 2x)}=2$

What is sum of the roots of the equation $(\sin x+\cos x)^{(1+\sin 2x)}=2\quad$ where $x\in[-2\pi,4\pi]$ ? We have $1+\sin 2x=\sin^2 x+\cos^2x+2\sin x\cos x=(\sin x+\cos x)^2$. so the equation is $$(\...
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Solve for $\theta$: $\tan(2\theta) \tan(\theta) =1$

I need to solve this without breaking down $\tan(2\theta)$. I have got this solution. But the answer in the book is different. $$ \begin{align*} \tan(2\theta) &= \cot(\theta) \\ \tan(2\theta) &...
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sin(kx+2$\pi$) = sin(kx)

I'm reviewing the sinusoidal wave and I cannot figure out why the second equality holds here. Can someone explain? I know sin(x)=sin(x+2$\pi$) but I don't see how sin(kx+2$\pi$) = sin(kx).
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1answer
32 views

Trajectory of a point

A point is moving in an orthogonal axis system $xOy$. On the X - axis the motion is described by the equation : $$ x(t) = A \cdot \sin(\omega\cdot t + \phi_1) $$ On the Y - axis the motion is ...
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Evaluate $\lim_{h \to 0} \frac{\sin(\frac{h}{2})-\frac{h}{2}}{h\sin(\frac{h}{2})}$ without l'Hospital

$$ \lim_{h \to 0} \frac{\sin(\frac{h}{2})-\frac{h}{2}}{h\sin(\frac{h}{2})} $$ I've worked the last few hours on this equation and still didnt find a way to evaluate it. The idea I had was to bound ...
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$\sin(t)=\sum_{n=-\infty}^{\infty}\frac{8.n.i-2}{(16.n^2-1).\pi}.e^{4.n.t.i} \Rightarrow \sum_{n=1}^{\infty}\frac{1}{16.n^2-1}?$

Knowing that $\sin(t)=\sum_{n=-\infty}^{\infty}\frac{8.n.i-2}{(16.n^2-1).\pi}.e^{4.n.t.i}$ in the interval $0\le t \le \pi/2$, how can I find the value of $\sum_{n=1}^{\infty}\frac{1}{16.n^2-1}$?
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Why it has extra square bracket with $dim$ in $[d(v_a, v_b)]^{dim}$?

I'm studying the paper and encounter this equation which $d(·,·)$ being described as Euclidean distance between two data points and $dim$ represents the dimensionality of the input data space. $$...
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1answer
38 views

How to add two arcs together if they overlap

I'm writing some code for various calculations with arcs and lines and need help with some math. I have list of arcs and need to add them together if they "overlap", here is the available ...
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38 views

What's with Trigonometry? [duplicate]

What's with trigonometry? When I learned it, we were told to just use a calculator, which does make my life easier, but I do want to know, how does one calculate a trigonometric function by hand. Is ...
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2answers
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Generalization of consequence of law of cosine

There is the immediate consequence of the law of cosine stating that when fixing two sidelengths of a triangle and increasing the third, the vertex angle opposite of the third side increases as well. ...
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1answer
64 views

$|\sin(\theta)|=\frac{2}{\pi}-\frac{4}{\pi} \cdot\sum_{m=1}^{\infty}\frac{\cos(2m\theta)}{4 m^2-1}\Rightarrow \sum_{m=1}^{\infty}\frac{1}{16m^2-1} =$?

By the equation $|\sin(\theta)|=\frac{2}{\pi}-\frac{4}{\pi} \cdot \sum_{m=1}^{\infty}\frac{\cos(2m\theta)}{4m^2-1}$, how can I get the value of $\sum_{m=1}^{\infty}\frac{1}{16m^2-1}$? What I tried: If ...
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Find coordinate of end point of a line?

How do I find the coordinates of the endpoint of a line with its length, starting position and angle relative to the x-axis? Thank you in advance.
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Prove that $\frac{1+(1-x)\sin\left(\frac{1}{x}\right)}{2}$ is onto $(0,1)$

I have been working and researching on the following problem. Find a continuous function defined on $(0,1]$ with range $(0,1)$. I found the function $\frac{1+(1-x)\sin\left(\frac{1}{x}\right)}{2}$. ...
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Finite expansion of sin$(\delta)$cos$(\beta-\delta)$ [closed]

In the finite expansion of sin$(\delta)$cos$(\beta-\delta)$ ($\beta<<1, \delta<<1$), what should we do with the big Os, i.e. $O(\delta^3)O((\beta-\delta)^2) $, and $O(\delta^3)+O((\beta-\...
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1answer
48 views

Inequality $\Gamma\left(\sin\left(\frac{1}{x}\right)\right)-\Gamma\left(\sin^2\left(\frac{1}{x}\right)\right)-x+x^2+\frac{1}{3}\geq 0$

Problem found with the help of Desmos and my imagination . Let $x\geq 1$ then we have : $$\Gamma\left(\sin\left(\frac{1}{x}\right)\right)-\Gamma\left(\sin^2\left(\frac{1}{x}\right)\right)-x+x^2+\frac{...
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Integral and hyperbolic/trigonometric substituition

I want to solve the integral $\int \frac{1}{x\sqrt{x^2-1}}\, dx$. I have thought to consider a tranformation of variables: $x=\cosh{t}\implies dx=\sinh{t}$ and so $\int \frac{1}{x\sqrt{x^2-1}}\, dx=\...
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5answers
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Prove $\frac{5i}{2+i}=1+2i$

Since $5i=5e^{i \frac{\pi}{2}}$ And $2+i=\sqrt{5}e^{ix_1}$ where $x_1=\arctan \frac{1}{2}$, we have $\frac{5i}{2+i}=\sqrt{5}[\cos(\frac{\pi}{2}-x_1)+i\sin(\frac{\pi}{2}-x_1)]$ Now from here how do I ...
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Equality that does not hold in general for Wolfram Alpha

Why for WolframAlpha $\frac{|\sin(\frac{x}{2})|}{|\cos(\frac{x}{2})|}=\sqrt{\frac{|(1-cos(x))|}{|(1+cos(x))|}}$ is not true $\forall x$? WolframAlpha In my opinion the equality holds since $\sin(\frac{...
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1answer
27 views

How to prove that $z(t) = z(t + T_0) \forall \theta$ for a periodic function

i have the following question about a proof: $z(t) = sin(4\pi t )+2cos(9\pi t+\theta)$ show that $z(t) = z(t + T_0) \forall \theta$ i dont know how to prove it. I know you should use $cos(9\pi t+\...
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4answers
95 views

How to find $ \lim_{x \to \frac{\pi}{2}}\frac{\cot^{2}x }{1-\sin x} $?

I tried to use cosec identities to solve, but I am having trouble especially with the cot identity. The closest I got to the answer is $$ \frac{1-\sin^{2}x}{(\sin^{2}x - \sin^{3}x)} $$ which I don't ...
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i can simplify the general form of cos(nx)?

I'm in doubt about that: the general form of cos(nx) is: $$\sum^{\lfloor \frac{n}{2} \rfloor}_{k=0}\binom{n}{2k}(-1)^{k}cos(x)^{n-2k}(1-cos(x)^{2})^{k}$$ i can expand $(1-cos(x)^{2})^{k}$ using the ...
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How to precisely convey (in math sentences) one or more symmetries (wrt polar axis, θ=π, orand the pole)? Be there way conclusively to determine non?

The online math textbook accompanying the Precalculus course I'm taking provides methods in the form of basic substitution tests to guarantee whether each symmetry necessarily holds and posits that if ...
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62 views

Area that the Goat can graze.

In the figure below, a goat is tied at the point $A$ with a rope of length $5$ metres. Find the total area goat can graze if it cannot enter the triangular region. My attempt: We have from $\Delta ...
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Ratio of area of wall to area of its shadow

A rectangular vertical wall runs east to west. The sun is $30$ degree east of south and its altitude is $60$ degree. Find the ratio of the area of the wall to the area of its shadow. My try: The ...
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25 views

A Trigonometric Identity in Boltzmann transport theory

I am trying to show to the identity below: \begin{equation} \cos\theta' = \sin\theta\sin\alpha\sin\phi+\cos\theta\cos\alpha \end{equation} The angles are given below: Anyone can shade some light about ...
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1answer
54 views

Considerations on a problem of trigonometry

I have this picture where the textbook of my students of 17 years old say to find the area of the quadrilater $AOBC$ (with $|AO|=|OB|=r$ the radius of the circle with the center in $O$), your ...
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1answer
51 views

Finding the volume of a solid of revolution when the function in question is an inverse trigonometric one

Here are two questions: Find the volume of the solid of revolution, generated by rotating the region bounded by the graph of $y = \arcsin x$ and the lines $x = 1$ and $y = 0$ through 2$\pi$ radians ...
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Divergence of the series $\dfrac{x^n}{ \sin ( \pi \alpha n) }$ for $0<x<1$

$0<x<1$ $u_0 \in \mathbb{N}$ and $u_0 \geq 2$ $u_{n+1}= u_n ^{ u_n}$ $\alpha= \sum_{n=0}^{+ \infty} \dfrac{1}{u_n}$ We want to prove that the series $\sum \dfrac{x^n}{ \sin ( \pi \alpha n) }$ ...
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Why we take domain of inverse of Cosine [0,π] instead of 0,-π [closed]

Why the inverse of cos is 0,π instead of 0,-π
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How to find this line length and targeted point coordinates based on other points?

First of all I'm beginner in "advanced" math. For this reason I don't know how to compute this problem. Consider we have a generic rectangle with width W and height H. Also, consider that ...
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1answer
26 views

Determine the cosine of the angles of the triangle whose vertices are $(2,-1,1)$, $(1,-3,-5)$, $(3,-4,-4)$

I found this exercise in the book "Introduction to Linear Algebra" by Serge Lang. I suppose it's expected to be done using the formula for the angle between two vectors: $$ \cos \theta = \...
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2answers
46 views

$y=-1$ in polar coordinates

Convert $y=-1$ from rectangular coordinates into polar; solve for $r$. Not sure how to do this one, can someone explain? Thank you.
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4answers
72 views

Maximum value of $ y = 3 \cos\alpha \sin\alpha + \sqrt 3 \sin^2\alpha$

Can somebody point me in the right direction in finding the maximum value of $y$ for a given value of $\alpha$ for the following trigonometric expression. $$ y = 3 \cos\alpha \sin\alpha + \sqrt 3 \sin^...
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3answers
52 views

How to find the range in $f(x)=\operatorname{arccsc} \left(\sqrt{2x-x^2}\right)$?

The problem is as follows: Let the real function: $$f(x)=\operatorname{arccsc} \left(\sqrt{2x-x^2}\right)$$ Find the range of that function. According to my workbook, the official answer to this ...
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1answer
48 views

An elementary proof that cos(x) in the required condition takes only 5 rational values

Assume that $x$ is a rational multiple of $π$ such is $\cos(x)$ is also rational. Then the number of values of $\cos(x)$ under the conditions is? I did read niven's proof of this, but it uses complex ...
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Not sure how to start the problem I know the format just unsure of which one to use

$\tan a =\frac{5}{12}$, $\pi<a<\frac{3\pi}{2}$, $\sin b=-\frac{1}{2}$, $\pi<b<\frac{3\pi}{2}$ Im totally lost on what to do any help would be appreciated

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