Tagged Questions

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

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Find the integer solutions for $n$ in $\sin\dfrac \pi {2n} + \cos\dfrac \pi {2n} = \dfrac{\sqrt n} 2$.

Let $n$ be a fixed positive integer such that $\sin\dfrac \pi {2n} + \cos\dfrac \pi {2n} = \dfrac{\sqrt n} 2$ then find the value of $n$. I have no clue how to do this sum. I couldn't even try it.
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Limit to infinity of trigonometry

\begin{align*}\lim_{n\rightarrow \infty}\frac{n\left(\left(1-\cos^2\frac{16}{n}\right)\sin\frac{16}{n}\right)^{1/3}}{4}=\lim_{n\rightarrow\infty}\frac{n\left(\sin^2\frac{16}{n}\sin\frac{16}{n}\right)^{...
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Overlapping area of two circle's crossing it's center i.e., length of overlapping is greater than r of the circle. Circle's has equal area.

Let there be two circular coasters of equal area (and negligible height). The purpose of is to find how far the two coasters need to be moved on top of each other such that the area of the overlapping ...
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Constant functions periodic?

I dont understand the meaning of this line in my book - " $\sin^2x + \cos^2x$ is periodic but the fundamental period is not defined. " Why is the period not defined? $F(x)$ is $1$ here so it is a ...
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How to maximize $\cos\theta$?

I have a question about maximizing $\cos\theta$. I have the equation $y=H\cos\theta$, where $H$ is the fixed height of a triangle. The problem asks me to maximize $\cos\theta$, but I have no idea ...
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A trigonometric problem when calculating distance to the boundary of a convex hull

Suppose we have a sphere and a point outside of the sphere. We denote the point outside as $v$ and the origin of the sphere as $x$. The convex hull of the sphere and $v$ should be like an ice cream ...
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Can Someone help me with my trigonometry rotation, formula? [on hold]

I've been working on some code for a game to make a hit box, this question is just about the math though. Basically I'm trying to rotate an X, Y point(i guess according to the game it's Z,X Not sure ...
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How to prove such a hyperbolic sine cosine related equality? [on hold]

$$\ln \left(\frac{\left(1+\sqrt{5}\right)^2 \left(2+\sqrt{5}\right)}{4}\right)=\text{arcsinh }(2)+2 \text{ arccsch }(2)$$
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Geometry and chemistry

There's a really obvious geometric reason why the cosine of the bond angle in graphite is $-1/2$: the stuff consists of sheets shaped like honeycombs. There's also a really obvious geometric reason ...
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Maximum and minimum of $f(x)=\cos(\sin(x))-\sin(\cos(x))$

Given the function: $$f(x)=\cos(\sin(x))-\sin(\cos(x))$$ it has absolute maxima at $x=(2k+1)\pi$ with $k=0,1,..N$ and relative maxima at $x=2k\pi$. It is not clear where are the minima. Putting the ...
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Plotting triangles based on a single point with distance and angle.

I'm tasked with creating an arrowhead within a pdf program. I have a single point with at $x=5.6$, $y=4$ this would be point A of my triangle I want to make the sides equal at $90$ degrees angles ...
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$\sin \alpha = \frac{3}{5}$ and $\cos \beta = -\frac{12}{13}$ . Find the values that $\cos(\alpha+\beta )$ can get.

$\sin \alpha = \frac{3}{5}$ and $\cos \beta = -\frac{12}{13}$ . Find the values that $\cos(\alpha+\beta )$ can get. Here $0<\alpha < \frac{\pi}{2}$ and $\frac{\pi}{2}<\beta<\pi$. Yes I ...
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How do you find the radius of an arc given arc length and height?

. Please excuse the poor drawing, but how would you go about solving this problem? Known: Arc length Height (I'm not sure what the proper term for this parameter is.) Unknown: Sagitta Chord ...
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Length of elliptical segment given starting and ending points and slope

I would like to represent the flight path of a turning aircraft with an ellipse. Initially, the baseline turn is 180 deg, with a constant radius. The speed of the aircraft is constant. During the ...
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How to solve $7.51\tan{\theta} - 2.656(\sec{\theta})^2=0$

I'm trying to solve $7.51\tan{\theta} - 2.656(\sec{\theta})^2=0$ and the way that it's been done in my notes is by somehow changing the equation to $7.51\tan{\theta} - 2.656(\tan{\theta})^2 - 2.656=0$ ...
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Proof of $\sum_{n=1}^{\infty}\frac1{n^3}\frac{\sinh\pi n\sqrt2-\sin\pi n\sqrt2}{{\cosh\pi n\sqrt2}-\cos\pi n\sqrt2}=\frac{\pi^3}{18\sqrt2}$

Show that $$\sum_{n=1}^{\infty}\frac{\sinh\big(\pi n\sqrt2\big)-\sin\big(\pi n\sqrt2\big)}{n^3\Big({\cosh\big(\pi n\sqrt2}\big)-\cos\big(\pi n\sqrt2\big)\Big)}=\frac{\pi^3}{18\sqrt2}$$ I have no ...
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Trigonometry Problem Solving

How can we estimate the height (h) of a castle surrounded by a moat, using the info below?
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Find all the angles $v$ between $-\pi$ and $\pi$

Find all the angles $v$ between $-\pi$ and $\pi$ such that $$-\sin(v)+ \sqrt3 \cos(v) = \sqrt2$$ The answer has to be in the form of: $\pi/2$ (it must include $\pi$) I have tried squaring but I get ...
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What maths would most likely have used for this game's horizontal bullet spread? Firing at 90° y causes the marks to line up perfectly.

While playing Doom, a game with a lot of mathematical techniques for various things, if I aim my x-as-well-as-y-spreading shotgun up at a 90° on the y view angle (x and y angles are used to look ...
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I was trying to find out the intervals where $\sin ^{-1}x > \cos ^{-1}x$

I was trying to find out the intervals where $\sin ^{-1}x > \cos ^{-1}x$ The easiest way was to just look at the graph and I found out that the region is $x \in ({1\over \sqrt{2}} , 1]$ But I ...
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Why does $\cos(x) + \cos(y) - \cos(x + y) = 0$ look like an ellipse?

The solution set of $\cos(x) + \cos(y) - \cos(x + y) = 0$ looks like an ellipse. Is it actually an ellipse, and if so, is there a way of writing down its equation (without any trig functions)? What ...
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Math precalculus/trig

Circle $O$ below has radius 1. Eight segment lengths are labeled with lowercase letters. Six of these equal a trigonometric function of $\theta$. Your answer to this problem should be a six letter ...
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Find maximum $2\sin 5x-3\cos x$.

Is it possible to find the maximum of $2\sin 5x-3\cos x$ without using calculus nor numerical methods? I suspect there is a way to play around with trig identities until the expression is only in ...
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What is the problem in my computation of $\sin 18^{\circ}$?

I needed to compute $\sin 18^{\circ}$. Now, these two relations hold for every $x$: $\cos 5x=16\cos^5x-20\cos^3x+5\cos x$ $\sin5x=16\sin^5x-20\sin^3x+5\sin x$, which can be easily proved using the ...
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Prove the following trigonometric result

If $\theta_1,\theta_2(0\leq\theta_1,\theta_2<2\pi)$ are two solutions of $\sin(\theta+\phi)=\frac{1}{2}\sin(2\phi)$, prove that \frac{\sin(\theta_1)+ \sin(\theta_2) }{ \cos(\theta_1)+ \cos(\...