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

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Interpolating a vector about an arc (Slerp)

In the following image, how can I solve for $k_0$? I know that $\mathbf v_1$ is a unit vector and $k_1 = \sin tω/\sin ω$.
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3answers
75 views

New very simple Golden Number Ratio PHI construction with Circle and Two Equal Segments of Circle Diameter. Is there prior art? Proofs?

Geogebra gives me the golden number PHI to fifteen decimal places for this simple construction illustrated below wherein the ratio of the blue line i to the red line h is PHI or 1.6180.... The golden ...
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2answers
41 views

Does $f(x) = -\sin(2x)$ have two integrals?

I found $\cos^2(x)$ and $\sin^2(x)$ which happily differ by the constant of $1$ though I've also found $\frac 12 \cos(2x)$, which both of the former diverge from by a sinusoidal function, what's wrong ...
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1answer
21 views

proof of $\sin(420º+\alpha) + \cos(60º+\alpha) = \sin(90º-\alpha)$?

I was trying to proof this using the right side, and I'm aware that $\cos (60 + \alpha) + \cos(60 + \alpha)$ it's what I'm really looking for but I can't find a way to proof it. \begin{align} \sin ...
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4answers
49 views

how to verify $\frac{\sin(x)\cos(x)}{\cos^2(x)-\sin^2(x)}=\frac{\tan(x)}{1-\tan^2(x))}$? [on hold]

How would I verifty the following trig identity? $$ \frac{\sin(x)\cos(x)}{\cos^2(x)-\sin^2(x)}=\frac{\tan(x)}{1-\tan^2(x)} $$ I am not sure how to start.
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3answers
37 views

Prove that the evolute of the tractrix $x=a(\cos t+\log \tan\frac{t}{2}),y=a\sin t$ is the catenary $y=a\cosh (\frac{x}{a})$

Prove that the evolute of the tractrix $x=a(\cos t+\log \tan\frac{t}{2}),y=a\sin t$ is the catenary $y=a\cosh (\frac{x}{a})$ Since evolute of a curve is the envelope of the normals of that curve.I ...
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2answers
41 views

Getting $\sin^2$ and $\cos^2$ values from $\sin^2 \alpha = \frac{1}{4} \cos^2 \alpha = \frac{1}{4}(1 - \sin^2 \alpha)$

How can you get from $$\sin^2 \alpha = \frac{1}{4} \cos^2 \alpha = \frac{1}{4}(1 - \sin^2 \alpha)$$ to $$\sin^2 \alpha = \frac{1}{5} \\or \\ \cos^2 \alpha = \frac{4}{5}.$$ Sorry, but I can't see ...
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22 views

Derivative atan2 of a function

I am not able to understand how to solve my doubt. I need to do the : $\frac{\partial}{\partial p} atan2({\cos(\alpha)},{\sin(\alpha)})$ I will compute $\cos(\alpha)$ and $\sin(\alpha)$ as: ...
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1answer
22 views

Convergence of a Fourier series to a point

Consider the function $f\left(x\right)=1+x$, $x \in \left[-\pi,\pi\right]$ I have calculated its Fourier series to be $$f\left(x\right)=1+2\sum^{\infty}_{n=0}\dfrac{\left(-1\right)^{n+1}}{n}\sin ...
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39 views

Finding sides using trigonometry.

I am currently studying for a maths exam and I came upon a question with this diagram With the question asking to solve all of the variables. No other information was given. How is this possible? ...
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3answers
40 views

Value of the given expression …

If $$y=\tan^{-1}\left(\sqrt{\dfrac{1+\cos x}{1-\cos x}}\right)$$ then value of $(2x+14y)^3-343$ is ? I reduced the equation as $y=\tan^{-1}\left(\dfrac{1+\cos x}{\sin x}\right)$ but I couldn't ...
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1answer
46 views

Why does $\sin\phi=r\frac{d\theta}{ds}$ and $\cos\phi=\frac{dr}{ds}?$

The relation between $p$ and $r$ where $p$ is the length of the perpendicular from the fixed point $O$ on the tangent to the curve at any point $P$ is called pedal equation of the curve. I want to ...
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1answer
15 views

Calculate edge of right triangle, two edges given

Given two coordinates of a right triangle's leg/cathetus' edges (x0,y0),(x1,y1) and the length of the other leg/cathetus (L). How do I calculate the coordinates of the remaining edge depending on ...
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0answers
28 views

Perimeter and Circumference of n-sided polygon

Given the sidelength, number of vertices and vertex angle in the polygon, how can one calculates the perimeter of an n-sided polygon that circumscribes a circle of radius r. And how can they use that ...
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0answers
14 views

Infinite sum of inverse trigonometric function [on hold]

$$\sum_{n=0}^{\infty} atan((acot(n^2+n+3))/(1+acot(n+1)acot(n+2)))$$
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3answers
41 views

How can I compute tan(.5*arctan(x))?

The plot for this function appears to be in the form of $\alpha*arctan(\beta*x)$ but I've no clue how to go about simplifying the expression.
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4answers
173 views

New very simple golden ratio construction incorporating a triangle, square, and pentagon all with sides of equal length. Is there any prior art?

Consider three regular polygons with 3, 4, and 5 sides wherein all the polygons have sides of equal length X throughout, as illustrated below. The ratio of the red line segment a to the blue line ...
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0answers
30 views

Cos(x/5)=(-1/3)? How many solutions for x in interval [-2π, 2π]?

I know that the all real solution is x=9.55316618 ± 10πn, 4.37254676 ± 10πn'. I know how to get x=9.55316618 ± 10πn, but the second part, 4.37254676 ± 10πn I don't know how to get. Can anyone show me ...
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1answer
36 views

Convergence of $\sum_{k \geq 1} e^{-tk} \cos kz$

I would like to find the convergence of the series $\sum_{k \geq 1} e^{-tk} \cos kz$. Clearly, this series converge in using the comparison test or the integral. How could I get an explicit function ...
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1answer
23 views

What is the full width of a peak of the function $F(X)=\frac{1+\cos((2N+1)πX)}{1+\cos(πX)}$

With $$1 + \cos \theta = 2 \cos^2 \frac{\theta}{2},$$ the function becomes $$f_n(x) = \left( \frac{\cos \frac{(2n+1)\pi x}{2}}{\cos \frac{\pi x}{2}} \right)^2.$$ It peaks at odd X integer values. ...
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1answer
22 views

if $f(x+3\pi)=f(x)$ where $f(x)=\cos{(nx)}\sin{(\frac{4}{n}x)}$ [on hold]

Let $n$ be integer number,if $$f(x)=\cos{(nx)}\sin{(\dfrac{4}{n}x)}$$ such $f(x+3\pi)=f(x),\forall x\in R$ Find $n$ since ...
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4answers
36 views

Finding the limits of a trig function

I have been struggling with finding the following limit: $$ \lim_{x\to \pi} \frac{\cos x + 1}{x - \pi} $$ Use of L'Hospital's rule is not permitted. Thanks
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0answers
7 views

How can I find the necessary speed and speed of rotation for a problem from a parametric equation?

I have been given the following questions for a project that I am currently working on: Questions 1 to 8 I have completed questions 1 through 6 but have no idea how to do questions 6 or 7 after ...
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0answers
26 views

solving equation invloving both algebraic and trigonometric terms

$$ x\sin(3)+3\sin(x)-xlog(3)+3log(x)=10$$ I need to know a method that finds all the possible values of x that satisfy the above equation.
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2answers
36 views

Express $\sin^3x$ in terms of cosines of multiples of $x$

I am studying complex numbers, and I have been trying to figure that out. Just not getting it. I keep getting $\frac{1}{-i8 (2\cos(3x) - 2\cos(x) - i4\sin(x))}$.
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52 views

About a geometric algorithm to compute $\sin$ based on the unit circle

In an old post I have found a user which claims to have a geometric algorithm to compute trigonometric  functions for an angle between $0^\circ$ and $90^\circ$ based on the unit circle. Here's the ...
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1answer
29 views

Can someone help me solve this word problem on vector and magnitude?

A river is 2000 feet wide and flowing at 6mph from north to south. A guy in a boat starts on the east shore and heads west at a normal paddling speed of 2mph. In what direction (measured clockwise ...
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50 views

Find nearest points on the circumference of a circle based on reference coordinates and centre coordinates given

For a programming purpose I've been asked to plot few points next to a point on a circular diagram. The only given values are the reference point coordinates and distance from/to of the new point to ...
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0answers
24 views

How to use Euler's formula to get the following identity

I'm reading a textbook and in the chapter on Euler's formula it is said that it's very useful for deriving all sorts of trigonometric identities, and the example given is: Where ||zθ|| = 1 I've ...
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19 views

Constant of proportionality for the following

Could somebody explain to me what is meant by the constant of proportionality and then show me how I would find it for the following polygons: Pentagon. Hexagon. Decagon. Kilogon. and Megagon. ...
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35 views

How to integrate $\cos2\pi\left(x+\frac{n}{x}\right)$

This is a follow up question of Integrate $\cos^2(\pi x)\cos^2(\frac{n\pi}{x})$. By using product to sum formula, this could be converted to question to integrate ...
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0answers
38 views

Pythagorean triplet triangle associated with Golden Ratio Construction. [on hold]

MoreGoldenPythagorusTriangle Just recognised simple connection by Astrophysics Math with a particular Pythagorean triplet proportion leads to an easy method to construct $ \varphi$ or its reciprocal. ...
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243 views

New, extremely simple golden ratio construction with two identical circles and line. Is there any prior art? [duplicate]

This question is different from a previously asked question (linked above) as this golden ratio construction only utilizes two circles and a line, and is thus far simpler than the golden ratio ...
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2answers
130 views

New Golden Ratio Construction with Two Adjacent Squares and Circle. Have you seen anything similar?

The below Golden Ratio Construction results in a ratio of PHI (1.6180...) between the blue line and red line, as found in Geogebra. This seems like a simple construction of the golden ratio, yet so ...
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42 views

About a curious nested radical representation for $\cos 1^\circ$

I have found the following nested radical representation. By using the triple angle formula for the cosine, $\cos 3\theta$, and making $\theta = 1^\circ$, we get the cubic equation $ 4x^3-3x = \cos ...
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2answers
22 views

Equivalence between trigonometric functions

could you help me to understand why: $\sin\left(x-\frac{2\pi}{3}\right)=-\cos\left(\frac{\pi}{6}-x\right)$ ? Thank you for your help.
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1answer
34 views

find the maximum of the function $f(x)=a+b\sqrt{2}\sin{x}+c\sin{2x}$

let $a,b,c\in R$,and such $a^2+b^2+c^2=100$, find the maximum value and minimum value of the function $$f(x)=a+b\sqrt{2}\sin{x}+c\sin{2x},0<x<\dfrac{\pi}{2}$$ Use Cauchy-Schwarz inequality?
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1answer
42 views

Easy trig problem. An impossible angle.

I was helping a friend and we encountered a particularly confusing problem. In triangle ABC, side a = 12, b = 16, and sinB = 2/3. What is the measure of angle A in degrees? The answer is 30, but I ...
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1answer
28 views

How to plot Trigonometric Functions in an Excel Spreadsheet?

I have been given the silly task of plotting some trigonometric functions in MS Excel. My task is to plot Sine, Cosine and Tangent on the same set of axes. Now, I get the Sine and Cosine waves ...
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66 views

What does the graph of $\sin^2 x$ look like?

When solving trig equations, sometimes it comes out as a hidden quadratic, like this: $2\sin^2x-5\sin x+2=0$ Obviously it is possible to factorise and solve for $\sin x$. I understand graph ...
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1answer
64 views

Have you seen this golden ratio construction before? Three squares (or just two) and circle. Geogebra gives PHI or 1.6180.. exactly

Note this golden ratio construction has been dramatically updated here with numerous golden harmonies: A Golden Ratio Symphony! Why so many golden ratios in a relatively simple golden ratio ...
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1answer
39 views

$t\mapsto\sin(tA)$ is continuous

How to show that $t\mapsto\sin(tA)$ is continuous for a real matrix $A\in Mat(n,n)$ Can I use trigonometric identity, $\sin y-\sin x=2\cos\left(\frac{x+y}{2}\right)\sin(y-x)$ but this holds ...
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3answers
54 views

If $\sin(t) = \frac{1}{2}$, why is $\sin (-t) = -\frac{1}{2}$

Given by an exercise in my book it is stated that: $\sin(t) = \frac{1}{2}$ Evaluate $\sin (-t)$ The given answer is $-\frac{1}{2}$. Why is that so, when sin is an even function?
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1answer
47 views

$\lim\limits_{i\mapsto \infty} \sum_{i=1}^\infty \sin(\theta_{i+1}-\theta_{i})$ = ? , where $\theta_{i}= \pi\sum_{j=0}^i \frac{1}{{(2)}^j}$

$\lim\limits_{i\mapsto \infty} \sum_{i=1}^\infty \sin(\theta_{i+1}-\theta_{i})$ = ? $\theta_{i}= \pi\sum_{j=0}^i \frac{1}{{(2)}^j} = \pi \left(2 - \frac{1}{2^{i}}\right)$ The limit ...
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1answer
32 views

Show that $\sinh(x)$ is strictly monotonic.

Is it enough to show that its derivative, $\cosh$, has no zeroes? (No $x$ satisfies $e^x=-e^{-x}$)
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68 views

Find the exact value of $2\left(\cos\frac{4\pi}{19}+\cos\frac{6\pi}{19}+\cos\frac{10\pi}{19}\right)$

$$S=2\left(\cos\frac{4\pi}{19}+\cos\frac{6\pi}{19}+\cos\frac{10\pi}{19}\right)$$ I've tried the sum of $2 \cos$ but can't find any solution. Please help!
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2answers
30 views

How to find intersections of sine and cosine functions with $X$ axis

I've been struggling with this question for a few days, because I've been able to find the said intersections, but based on suppositions, rather than on mathematical process. For example, if I have ...
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0answers
31 views

Prove trigonometric inequality with sin

Let $ n\in \mathbb{N}^{*},x\in \mathbb{R} $. Prove that $ sin^{2}(x)\cdot sin^{2}(2x)\cdot ...\cdot sin^{2}(2^{n}x)\leq \left ( \frac{3}{4} \right )^{n},\forall x\in \mathbb{R} $. The only result I ...
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3answers
36 views

sum of all Distinct solution of the equation $ \sqrt{3}\sec x+\csc x+2(\tan x-\cot x) = 0\;,$

The sum of all Distinct solution of the equation $\displaystyle \sqrt{3}\sec x+\csc x+2(\tan x-\cot x) = 0\;,$ Where $x\in (-\pi,\pi)$ and $\displaystyle x\neq 0,\neq \frac{\pi}{2}.$ ...
2
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4answers
129 views

Extended $\lim_{x \rightarrow 0}{\frac{\sin(x)}{x}} = 1$ limit law?

So I've learned that $\lim_{x \rightarrow 0}{\frac{\sin(x)}{x}} = 1$ is true and the following picture really helped me get an intuitive feel for why that is I have been told that this limit is ...