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

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2
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1answer
102 views

Why is $e^{g(x)} = \pi$ where $g(x)$ is holomorphic in Weierstrass factorization of sine function?

Why is $e^{g(x)} = \pi$ where $g(x)$ is holomorphic in Weierstrass factorization of sine function? I just can't get why it's true.
3
votes
1answer
65 views

Trig and algebra problem: Finding sides of a triangle

Let $ABC$ be a triangle such that $\angle ACB = \pi/6$ and let $a,b,c$ denote the lengths of the sides opposite to $A,B,C$, respectively. What are the value(s) of x for which $a = x^2 + x + 1, b = ...
0
votes
1answer
54 views

Identifying degrees and radians

I have the following problem : If $\sec(1.4) = x$, find the value of $\csc(2\tan^{-1}x)$. (A) $0.33$ (B) $0.87$ (C) $1.00$ (D) $1.06$ (E) $3.03$ I we take the $1.4$ as degrees, we get option ...
0
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2answers
60 views

Given three plots of trigonometric functions, find an expression for each function [closed]

Considering the three graphs in figure 1 showing trigonometric functions, $f(x)$, $g(x)$ and $h(x)$. Using these graphs, write the expression for each function.
6
votes
1answer
157 views

Solve $\sin(x)+2\sin(x)\cos(x)=\pi/4$

Is it possible to solve (not approximate) the following trigonometric equation by hand? $$\sin(x)+2\sin(x)\cos(x)=\pi/4.$$
2
votes
1answer
470 views

Cone shaped related rates of change question

A container is in the shape of a cone of semi-vertical angle $30^\circ $, with it's vertex downwards. Liquid flows into the container at ${{\sqrt {3\pi } } \over 4}{\rm{ }}c{m^{^3}}/s$ At the ...
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votes
4answers
3k views

Learning trigonometry on my own.

I have been self teaching myself math beginning with a grade 10 level for a while now and need learn trigonometry from near scratch. I am seeking both books and perhaps lectures on trigonometry and ...
0
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1answer
157 views

Solving a trigonometric inequality

I am solving this inequality for $\theta$ $$\cos \theta > -\frac{\rho}{\sqrt{1-\rho^2}} \sin \theta$$ with $\rho \in (-1,1)$ given, when trying to integrate a function under polar coordinates and ...
0
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2answers
70 views

the period of a trigonometric function

I'm trying to solve a differential equation which is : $$y'(t)-4y(t) = \cos(3t)$$ Resolution of the equation without the second membre $y'(t)-4y(t)=0$ has as solution $ y_s(t)=ke^{4t} $ with ...
2
votes
5answers
4k views

The hypotenuse of a right angle triangle measures 12 cm. What size angles would produce maximum…

The hypotenuse of a right angle triangle measures 12 cm. What size angles would produce the maximum perimeter? I got to point where I take the derivative and get $12(\cos\theta-\sin\theta)=0$, not ...
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0answers
145 views

Getting the angle between three points

So I have this psuedo code here (converted from c# to show you better) ...
4
votes
1answer
93 views

Optimal rotation to align a circle with external points

I have a circle $C$ with radius $r$ and a set of finite points $P=\left \{ p_1,p_2,\ldots,p_n \right \}$ are identified external to the circle $C$. These points may lie on the exterior or the interior ...
0
votes
2answers
202 views

Real and Imaginary Parts of $\frac{\cos(z)}{(1-e^{ix})}$

Find $$\mathrm{Re}\bigg(\frac{\cos(x+iy)}{(1-e^{ix})}\bigg)$$ and $$\mathrm{Im}\bigg(\frac{\cos(x+iy)}{(1-e^{ix})}\bigg)$$ Please help I've been trying for some time now...
1
vote
1answer
81 views

finding Length of a diagonal

Given Quadrilateral ABCD in such that $AB<BC<CD$ creating increasing arithmetic progression with sum of $27$ cm. $\measuredangle BCD=60^{0}$. the diagonal $BD=\sqrt{133}$ cm, and it divided ...
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3answers
4k views

Calculating circle radius from two points and arc length

For a simulation I want to convert between different kind of set point profiles with one being set points based on steering angles and one being based on circle radius. I have 2 way points the ...
1
vote
2answers
191 views

What is the best way to solve equations with trig functions

I usually use guessing to solve equations with trig functions. Yesterday, I came across an equation that I couldn't really write it in a helpful form to guess. My question is, how can I solve equation ...
1
vote
1answer
53 views

Angle consistency between vectors in N dimensions

I am trying to understand how rotations work in higher dimensions. Let us assume we have a set of points $p_i\in P$ in $N$ dimensions, related to another set of points $q_i \in Q$ by a rotation $R$. ...
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5answers
985 views

Show that $\sin^{-1}( x) =\tan^{-1}(x/\sqrt{1-x^2})$ for $|x| <1$

Show that $\sin^{-1}( x) = \tan^{-1}(x/\sqrt{1-x^2})$ for $|x| <1$ I have got as far as knowing that values of $\sin^{-1} x$ are only defined when x lies in the set [-1, 1]. and that for any ...
4
votes
1answer
145 views

eye vision problem

Imagine that the smallest letter that ken can read on the Snellen Eye chart is 3 inches tall. What is Ken's vision, using 20/XX notation? I have a question about eye vision, but I have no idea how ...
2
votes
3answers
119 views

Fractional Trigonometric Integrands

$$∫\frac{a\sin x+b\cos x+c}{d\sin x+e\cos x+f}dx$$ $$∫\frac{a\sin x+b\cos x}{c\sin x+d\cos x}dx$$ $$∫\frac{dx}{a\sin x+\cos x}$$ What are the relations between the numerator in the denominator, and ...
3
votes
2answers
237 views

How to find the solutions $x$ of $ 2\sin{11^{\circ}}\sin{71^{\circ}}\sin{(x^{\circ}+30^{\circ})}=\sin{2013^{\circ}}\sin{210^{\circ}}$

Let $$2\sin{11^{\circ}}\sin{71^{\circ}}\sin{(x^{\circ}+30^{\circ})}=\sin{2013^{\circ}}\sin{210^{\circ}}$$ where $90^{\circ}<x<180^{\circ}$. My idea: ...
0
votes
1answer
21 views

What is Angle(A,b) about something.

I was reading a paper and came through a notation saying .... Angle = Angle(A,B) about C. Can anybody tell me what exactly it means. Thnaks, Harsha
0
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2answers
172 views

Need “up” vector to calculate distance from a focal plane given world coordinates (SOLVED)

I have a RGB image, and for each pixel in the image I also have its real world coordinate. I also have the location (real world coordinate) yaw, pitch and roll of the camera. I am trying to produce ...
0
votes
2answers
572 views

Trapezoid rule over trigonometric polynomials

The question is regarding trapezoid rule applied on trigonometric polynomials Here is the question Show that the composite trapezoid rule over an equidistant partitioning with interval size $h = ...
0
votes
3answers
189 views

Differentiate $y = {(x + 2)^3}{(1 - \sin 2x)^2}{(1 + \tan x)^3}$

I haven't got very far in attempting this: $\eqalign{ & y = {(x + 2)^3}{(1 - \sin 2x)^2}{(1 + \tan x)^3} \cr & y = {\left( {(x + 2)(1 + \tan x)} \right)^3}{(1 - \sin 2x)^2} \cr} $ I'm ...
10
votes
2answers
354 views

Inequality $\sum_{1\le k\le n}\frac{\sin kx}{k}\ge 0$

Show the following inequality for any $x\in [0, \pi]$ and $n\in \mathbb{N}^*$, $$ \sum_{1\le k\le n}\frac{\sin kx}{k}\ge 0. $$ I have this question a very long time ago from a book or magazine but I ...
1
vote
1answer
109 views

Calculate points(x, y) within an arc

I am trying to draw lines from the center of a circle to points (x, y) in the circumference. To calculate this the angle is used. I need to render points in between two angles. E.g. Angle 0 to angle ...
3
votes
3answers
556 views

How do you determine the local extrema points for $y=\sqrt{3}\cos(3x)+\sin(3x)$

$$y=\sqrt{3}\cos(3x)+\sin(3x); 0\le{x}\le{\frac{2\pi}{3}}$$ I know that the local extrema can be determined by using the first derivative test. I took the derivative of $y$ and got $$y'= ...
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vote
1answer
44 views

How to get to these steps?

I found this question here (I recommend that you read the question and the highest-voted answer there) How to solve for $x$ in $x(x^3+\sin x \cos x)-\sin^2 x =0$? and the math below is an answer. I ...
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vote
4answers
615 views

$\lim_{x\to 0} x^3/\tan^3(2x)$

$$\lim_{x\to 0}\frac{x^3}{\tan^3(2x)} $$ My textbook has an answering of $\frac{1}{8}$ and I'm quite confused on how they got that. Only thing that I could see to get an $8$ would be $2^3$ from ...
1
vote
1answer
100 views

Calculating circle properties.

How can I incrementally calculate the angle from angle 0 and the point (x, y) in a circumference path if I have the center of the circle coordinates and the radius of the circle. I have 127 segments ...
1
vote
1answer
234 views

Prove trigonometric inequality

Could anyone help me prove the following inequality for $x>0$$$x(2+\cos x)>3\sin x$$ If you could just show me the first few steps, that would be great.
2
votes
2answers
55 views

Simplifying $\prod_{k=0}^n \cos(2^{-k})$

A student of mine has trouble with the following, and so do I. The solution should be easy since it has been ask to première S students (equivalent to American 11th grade I guess). The question is ...
4
votes
2answers
487 views

How to find the maximum diagonal length inside a dodecahedron?

I am trying to find the maximum length of a diagonal inside a dodecahedron with a side length of $2.319914107\times10^{89}$ meters. I am not sure if any other information than that is needed, if it ...
11
votes
3answers
219 views

Could we show $1-(x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dots)^2=(1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}- \dots)^2$ if we didn't know about Taylor Expansion?

Suppose that humanity haven't discovered Taylor Series Expansion of trigonometric functions or of any function that would help us on this. Which means we are not allowed to replace the given infinite ...
0
votes
1answer
136 views

Prove the inequality $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$

Let a, b, c be the three side lengths of a triangle. Prove that $$\frac{a}{b+c-a}+\frac{b}{a+c-b}+\frac{c}{a+b-c}\geq 3$$ Under what conditions is equality obtained?
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2answers
534 views

How do you solve $z^4 = 2(1+i\sqrt{3})$

Solve $z^4 = 2(1+i\sqrt{3})$ in the form $r(\cos\alpha+i\sin\alpha)$ where $r>0$ and $0\le\alpha<2\pi$ I know you have to find $\arctan(\frac{\sqrt{3}}{1})=\frac{\pi}{3}$ and that is $\alpha$? ...
1
vote
1answer
76 views

Euclidean triangle. Does this one exist

Does $\exists$ a Euclidean triangle $ABC$ with $\sin(A) : \sin(B) : \sin(C) = \frac{1}{4} : \frac{1}{3} : \frac{1}{2}$?
2
votes
1answer
6k views

Time Average of Cosine squared function

I've carried out the steps for the time average for $\cos^2x$ for limits $0$ to $T$. I've gotten : $\frac{1}{T}\left[\frac{1}{2}[T+\frac{1}{4}\sin2T\right]$ I'm trying to find the average over a ...
4
votes
1answer
422 views

Largest Quadrilateral from a Set of Points

I posted the below on StackOverflow but was directed here as this may be more mathematical problem but I was looking to implement an algorithm.... I have a discrete set of points. From this set of ...
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1answer
591 views

Time average of $\cos^2 x$ function [closed]

How do I find the time average of $\cos^2(3-wt)$ ?
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2answers
85 views

Turning points on $2\sin x - x$

I'm self teaching and doing a book exercise which asks: "Considering only positive values of x, locate the first two turning points on the curve $2\sin x - x$ and determine whether they are maximum or ...
5
votes
3answers
1k views

Simplifying $\sin(2\tan^{-1} x)$

I've been working on this for a while. The answer in the book is $\frac{2x}{x^2 + 1}$ Here's my workings: $\sin(2\tan^{-1} x)$ Let $\alpha = \tan^{-1}x \Rightarrow \tan \alpha = x$ $\sin(2\alpha) = ...
4
votes
1answer
267 views

How can I calculate the angle of a slice of an ellipse?

I'm attempting to draw a pie-chart programmatically, using an ellipse instead of a circle, but I'm having trouble calculating the correct angles for the slices. If it were a circle, I could use the ...
3
votes
1answer
130 views

Find the maximum value of $T=\frac{2}{3}(\cos 2A-\cos 2B)-\tan\frac{C}{2}$

Let $ABC$ be a triangle. Find the maximum value of $$T=\frac{2}{3}(\cos 2A-\cos 2B)-\tan\frac{C}{2}$$ Please give me some hints. I don't know where to start Thanks
0
votes
1answer
203 views

Coordinates of all 'N' points, equidistant from each other , on a circle of radius 'R' whose center is (h,v) from the origin?

How would I calculate the coordinates of all 'n points' equidistant from each other on a circle of radius r and the center coordinates of (h,v) from the origin .
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3answers
450 views

Broken Calculator: only certain unary functions work.

I have run into a challenge on Codecademy.com that has me absolutely bewildered. I'm sure I'm just overlooking an obvious solution, but I've been scouring tables of trigonometric and logarithmic ...
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2answers
155 views

Proving that $\frac {\cos(π + x)\cos(-x)}{\cos(π-x)(\frac{π}{2}+x)} = \cot^2(x)$

Prove: $$\frac {\cos(\pi + x)\cos(-x)}{\cos(\pi - x)\cos(\frac{\pi}{2}+x)} = \cot^2(x)$$ I tried to solve the left hand side but got the answer as $-\cot(x)$ instead.
0
votes
1answer
74 views

How to find a new point on rectangle based on an known point on the same?

I have rotated a rectangle a certain amount of degree and got the point(x,y)=(130,40) which was previously (152,60). Now i want to find the x,y(marked as red) value at another location based on the ...
0
votes
1answer
53 views

Prove that $x^r$ with $r=a+ib$ or $r=a-ib$ defines real solutions

Prove that $x^r$ with $r=a+ib$ or $r=a-ib$ defines real solutions in terms of trigonometric functions with argument $\ln x$ multiplied by exponential function $y(x)=x^{(a+ib)x}$ or $y(x)=x^{(a-ib)x}$ ...