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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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0answers
14 views

Stuck with a possibly impossible trigonometry question

I need to find the length of the arc between Y1 and Z1 in the image below. If you can even get me to the value of Y, then that will work. I appreciate the drawing may be crude, but imagine that Y and ...
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2answers
39 views

Calculating exact values of “weird” functions like arcsin(sin 100)

This is pretty much the last thing I need to know for now. Tasks (calculate): $\arccos{(\cos{12})}$ $\arctan{(\tan{\sqrt{5}})}$ $\arcsin{(\sin{100})}$ Answers: $4\pi-12$ $\sqrt{5}-\pi$ $100-32\pi$ ...
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3answers
33 views

Calculating inverse trig expressions like cos(arctan -2)

I have some problems "connecting dots". All feedback is welcomed and really, really helpful! :) Task 1: calculate $\quad \tan{(\arcsin{(-\frac{3}{4}}))}$ Solution: $\tan{(\arcsin{-\frac{3}{4}})} = ...
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3answers
35 views

Cannot find angle for trigonometry problem

A right angle triangle is entrapped within a circle. The triangle entraps within it a circle of its own. The ratio between the big radius and the little radius is $\frac{13}{4}$. What are the angles ...
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1answer
25 views

How to represent inverse trig function as another inverse trig function?

I'm struggling with this one. How can I respesent some inverse function as another inverse function? Is it possible to represent let's say $\arctan(-2)$ in terms of $\arccos(...)$, $\arcsin(...)$, $\...
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2answers
28 views

Proving Trigonometry Identities $\tan(\frac{1}{2}x+45)+\cot(\frac{1}{2}x+45)=2\sec x$

How do i prove that $\tan(\frac{1}{2}x+45)+\cot(\frac{1}{2}x+45)=2\sec x$? I managed to arrive at $$\frac{2 \sec^2 (\frac{1}{2}x)}{1- \tan^2 (\frac{1}{2}x)}$$ but I got stuck here. Any help is ...
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5answers
33 views

How to solve inverse trig. equations like sin(arctan 2)?

These are some of the tasks I am supposed to be prepared for. I have no idea where to begin when solving them. Below I present what I already know regarding the subject and what I have problems with. ...
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1answer
15 views

Need help showing Riemann's Functional equation for negative numbers and complex numbers

Riemann's Functional equation: $\zeta(-z)$=${-2*z!\over(2\pi)^{z+1}}$$sin({\pi z\over2})$$\zeta(z+1)$This formulas expresses $\zeta(-z)$ in terms of $\zeta(z+1)$ Note: I read that the author said, ...
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1answer
67 views

Integrating $\int x \csc^2(\,\ln(x)\,)\,dx$

What is this integral? $$\int x \csc^2(\,\ln(x)\,)\,dx$$ I haven't found any solution yet. It's the same as this one $$\int \frac{x}{\sin^2(\,\ln x\,)}\,dx$$ So, how can I understand this ...
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1answer
30 views

Computing $\sum_{k=0}^n{2n\choose 2k}(-1)^ksin^{2k}\theta\ cos^{2n-2k}\theta$ using Euler's formula

Compute the following sum by using Euler's formula, $ e^{i \theta} = \cos \theta + i \sin \theta$, $$cos^{2n}\theta-{2n\choose 2}cos^{2n-2}\theta\ sin^2\theta\ +...+(-1)^{n-1}{2n\choose 2n-2}cos^2\...
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3answers
49 views

Using ratiotest to check convergence of $\sum \sin(\frac{1}{n!})$.

I'm preparing a class on the convergence of series and how to check it using de ratiotest. One of the exercises asks to determine the convergence of the series $$\sum \sin(\frac{1}{n!})$$ using the ...
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0answers
20 views

Show $\langle x-Px,y-Px\rangle \leq 0$ for $Px$ the projection of $x$ onto $C$ convex, and any $y \in C$.

So far I did the following: Write $x=w+Px$ for $w$ st. $Pw=0$. Then $\langle x-Px,y-Px\rangle=\langle w,y-Px \rangle$. From here I can intuitively see that is non-positive since 'the arrow' from $Px$ ...
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2answers
48 views
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1answer
195 views

Showing that $\cos^p{\Theta} \le \cos{p\Theta}$, for $0<\Theta<\frac{\pi}{2}$ and $0<p<1$, by analyzing $f(\theta)=\frac{\cos^p\Theta}{\cos p\Theta}$

Given $0 < \Theta < \frac{\pi}{2}$ and $0 < p < 1$, show that $$\cos^p{\Theta} \le \cos{p\Theta}$$ Can you please check if my proof is correct? Would also love to know if there're other ...
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2answers
543 views

Tracking the movement of the watch hand

I love watches, and I had an idea for a weird kind of watch movement (all of the stuff that moves the hands). It is made up of a a central wheel, with one of the hands connected to it (in this case, ...
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2answers
47 views

trig problem needs help plss [on hold]

A quarter circular sector or is removed from a circle and the remainder is folded into a cone by connecting the cut edges. When viewed from the side, what is the angle at the apex of the cone?
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1answer
41 views

Existence of basis of solution

Show that the function $\cos(2x) -2\cos^2(x)$ and $\cos(2x)- 2 \sin^2(x)$ cannot be a basis of solution of any Differential Equation with second order.
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3answers
49 views

Help explaining the simplification of an integral

I am trying to understand the following steps my teacher did in class (from top to bottom). I tried to look up different trigonometric identities but couldn't figure out where the arrival of cosine ...
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0answers
39 views

Sum of Cosine Waves

Plot two waves on excel: $\cos(f_1)$ and $\cos(f_2)$ and their sum $[\cos(f_1) + \cos(f_2)]$. The frequencies of the waves are $f_1= 2\pi*f$ and $f_2= 2 \pi*f*(1.1)$. Assume $f=1$ My question is how ...
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1answer
229 views

Atypical way to find angle between unit vectors: $\theta = 2 \sin^{-1}\left(\frac{1}{2}\left\|\hat{A}-\hat{B}\right\|\right)$

At my work, I have come across code with the following way of calculating the angle between two vectors. $$\theta = 2 \sin^{-1}\left(\frac{1}{2}\left\|\hat{A}-\hat{B} \right\|\right)$$ (Note the ...
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2answers
53 views

Geometric justification of a rotation matrix

From S.L Linear Algebra: We can define a rotation in terms of matrices. Indeed, we call a linear map $L: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ a rotation if its associated matrix can be ...
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2answers
30 views

Number of solutions to a given equation

Any hints on how to proceed further?
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0answers
41 views

Roots of equation involving both polynomials and trigonometric functions

I have to determine for which values of $x$ that the velocity vector is orthogonal to the acceleration vector, the position is given by: $(3 \cos(t), - \sin(3t), 2t^3 - t^2)$, I then use that $u \...
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1answer
37 views

Find number of possible values satisfying $\cos \left(\pi\sqrt{x-4}\right)\cos \left(\pi\sqrt{x}\right)=1$ [on hold]

Find number of possible values satisfying the equation: $$\cos \left(\pi\sqrt{x-4}\right)\cos \left(\pi\sqrt{x}\right)=1$$ We are required to find the number of possible solutions for $x\in\Bbb R$. ...
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3answers
37 views

Is this a True Property of the Lemniscate of Bernoulli?

I am trying to figure out if the following is true: Take the Lemniscate of Bernoulli (a plane curve defined from two given points F1 and F2, known as foci, at distance 2a from each other as the locus ...
3
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1answer
37 views

Degree choice in improper integrals resulting in trigonometric functions

people. I have a question regarding the following improper integral, and others like it: $$\int_{-\infty}^\infty \frac{dx}{1+x^2}$$ The end result of that are the two limits: $$\lim_{a\to -\infty} \...
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1answer
40 views

Simplifying $\csc^2(180^\circ + \theta ) - \sin^2(180^\circ - \theta) -\cos^2(-\theta)$ [on hold]

I am having some trouble answering this problem Simplify: $$\csc^2(180^\circ + \theta ) - \sin^2(180^\circ - \theta) -\cos^2(-\theta)$$ The answer that my textbook says is $$\csc^2(\theta) - \sin^...
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1answer
160 views

Can all trigonometric expressions be written in terms of sine and cosine?

I know that sine and cosine can be rewritten in terms of the real and complex parts of the exponential function as a result of Euler's formula. My question is, can every trigonometric expression be ...
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2answers
312 views

How do we really get the angle of a vector from the components?

Usually when people discuss getting the polar form of a vector $v$, they present the following two formulas: $$\text{Magnitude}(v) = \sqrt{x^2 + y^2}$$ $$\text{Angle}(v) = \arctan \left(\frac{y}{x} \...
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1answer
27 views

tilting a disc in 3d space - need help

Lets imagine you have a disc like a CD. Then you take that CD and rest it flat on a desk. Now you tilt the disc left to right and forward to back while touching the desk with 1 point on the edge of ...
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1answer
14 views

Fourier sine series simplification

I'm having trouble simplifying a Fourier sine expansion for the following function: $$f(x) = \max\{{\frac{\pi}{2}, x}\}$$ on the interval of $[0,\pi]$. Since we're doing a sine series then $a_n = 0$ ...
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2answers
50 views

$\sin(x)+\cos(x)+\sin(2x)>1$ [on hold]

I am having trouble solving this problem http://tinypic.com/view.php?pic=vsl5s5&s=9 In our books, we have solutions in the back and solutions for $x$ are different from mine. I get solutions ...
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1answer
40 views

Showing $\frac14 (1 -\cos 2\theta)^2 = \frac18(3 -5\cos 2\theta)$

I have an example which I have: $$D = \frac{2}{\sin^4\theta} \tag{1}$$ Which in the notes goes to $$D = \frac{2}{\frac14 (1 -\cos 2\theta)^2} \tag{2}$$ I understand that part.. but the next part ...
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2answers
422 views

How do I construct this triangle [on hold]

I was trying to draw the following triangle in latex tikz and I just could not find a way to do it with respect to the given conditions. Is it possible to ...
2
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1answer
38 views

How to solve this vector problem involving more than one unknown?

This is an exercise I came across while tutoring high school physics. I am posting this as an "answer my own question." Kyle suspends a 12340 N moose from two trees as shown below. What is the ...
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3answers
44 views

Help with complex numbers

Algebraic form of the $z^3 = (3 + i)^6$. Can someone help me to solve this? My answer is $z = 8 + 6i$, but I'm not sure that this is true.
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6answers
49 views

Find exact value of tan when given cos

Given $\cos30 = \frac{\sqrt3}{2}$ use trigonometric identities to find the exact value of $\tan\frac{\pi}{3}$ I understand that $\cos30 = \frac{\sqrt3}{2}$ from the standard trig values chart and I ...
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1answer
25 views

on the inverse of trigonometric or/ and hyperbolic functions

If we want to find, say, the inverse $\tan$ function, $\tan^{-1}$, in terms of (complex) logarithm function we start with the equation $z=\tan w =\frac{\sin w}{\cos w}=\frac{1}{i}\frac{e^{iw}-e^{-iw}}{...
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2answers
19 views

Get leg length depending on another leg knowing the perimeter of a rectangle triangle [on hold]

Knowing the perimeter of the triangle, how can i find the side (leg) b in function of the leg a and it's perimeter p.
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0answers
15 views

How to find the near close aspect ratio of a billboard from a distance photo?

I have a photo of a billboard. I am trying to solve this problem of finding out the aspect ratio of this billboard. What is known to me is nothing more than this photo. Can someone help me with ...
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3answers
42 views

Intriguing geometry problem regarding isogonal lines

A line $r$ contains the points $A,B,C,D$ in this order. Let $P\notin r$ such that $$\angle APB=\angle CPD$$ Denote furthermore by $G$ the intersection of the angle bisector of $\angle APD$ and $r$. ...
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0answers
9 views

Get sinus of an angle from a distance x and two circumferences of different radius

How is possible to get these formulas of sinus from this image? https://imgur.com/a/8MqyRFb I have been trying different things but they didn't work out.
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4answers
55 views

Trigonometric equation $ \cos{x} + \cos{\sqrt{2}x} = 2$

I can not find a good way to solve this rather simple-looking equation. $ \cos{x} + \cos{\sqrt{2}x} = 2$ I can see that 0 is a solution, but is there a good way of solving it for all the potential ...
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1answer
33 views

Expressing, in terms of real radicals, the trigonometric functions generated by cubic equations with integer coefficients

When solving a cubic equation, one might have to use trigonometric functions. In some cases, these trigonometric functions can be expressed in terms of real radicals. The goal is to find all the ...
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2answers
30 views

Is there always exactly one solution to $a \cos\left(\frac{x}{2}\right)- b \sin\left(\frac{x}{2}\right) = 0$ in the interval $0\leq x\leq 2\pi$?

I have a probably simple question. If I have an equation like \begin{align} a \cos \left(\frac{x}{2}\right) - b \sin \left(\frac{x}{2}\right) &= 0\\ \frac ab \cos \left(\frac{x}{2}\right) - \sin \...
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0answers
17 views

Inverse trigonometric formula logic?

What is the logic behind adding/subtracting pi in the above formulas? I have heard from people that graphs can help but I'm not sure how.
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3answers
77 views

How do you integrate $\int \frac{\cos(4x)}{\cos(x)}dx$?

I tried using trigonometric formulas for turning it into 2$\int \frac{\cos^2(2x)}{\cos(x)}dx - \int \frac{1}{\cos(x)}dx$ and can solve the second one, but still no idea of how to proceed with $\int \...
6
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4answers
83 views

Given a chord, how do I find the ellipse?

It will explain my use case at the end, in case I am approaching this wrong, but I will start with the math question. Given: a point $\rm P$ on an ellipse; the slope of the tangent (or normal) to ...
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1answer
26 views

Worst-Case Addition to Smallest Enclosing Circle

Imagine you have a convex polygon $p_1$ and put a smallest enclosing circle $SEC_1$ around it, the lengths of each side of the polygon $l_i$ can be anything and don't have to equal each other. Now, ...
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2answers
30 views

Find the x intercepts of $f(x)=\tan\left(x\right)-\tan\left(4x\right)$

I'm having serious trouble finding the x-intercepts of this funtion $$f(x)=\tan(x)-\tan(4x)$$ I'm unable to solve it for zero. What am I missing here?