# 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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### How do I find the exact value of $\cos\frac{\pi}{12}\cos\frac{5\pi}{12}\cos\frac{7\pi}{12}\cos\frac{11\pi}{12}$?

I know that $\cos(6\phi)\equiv32c^6-48c^4+18c^2-1$ where $c=\cos\phi$. I also know that when $\cos(6\phi)=0$, then $\phi=\frac{k\pi}{12}$ ($k = 1,3,5,7,9,11$). How do I find the exact value of: ...
268 views

### Intersection of a point and absolute value function contained within a circle

I'm attempting some crazy ideas while programming a game and ran into the following math problem that has been bugging me for a few days: Given a unit circle and a random point $P$ within the circle, ...
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### Sum : $\sum \sin \left( \frac{(2\lfloor \sqrt{kn} \rfloor +1)\pi}{2n} \right)$.

Calculate : $$\sum_{k=1}^{n-1} \sin \left( \frac{(2\lfloor \sqrt{kn} \rfloor +1)\pi}{2n} \right).$$
322 views

### Get $x,y$ coord of a rectangle when rotation is $0$

I have this square div in the page. I need to get the $x,y$ coords (left/top) of the box based on rotation 0 even when the box is rotated at a different angle. Is there a formula to calculate the the ...
122 views

### Euler's Basel problem continued… $\zeta(2n)$ expressed in terms of $sinc$?

I have to make a brief intro before comming to my question. To approach the famous Basel problem Euler starts with the $sinc$ function \begin{align}\frac{\sin(x)}{x} = 1 - \frac{x^2}{3!} + ...
308 views

### New x coordinate of a rotated line

I need help finding the equation to find $x$ I work in GIS and I'm working on a script that uses the new x coordinate of a rotated line. I havent work with trigonometry in a long long time so I ...
397 views

### Angular distribution of lines passing through two squares.

Let's say I've got two squares with side length $d$ that are held parallel at a distance $m$ apart. Suppose that particles are randomly falling from above in such a way that the polar angle ...
134 views

### Explanation of where this trig identity comes from

I'm working on a problem but it's been a while since I last saw trig identities so I'd love some help or being pointed in the right direction. Basically, I'd like to understand where this identity ...
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### Location of roots of certain transcendental equations.

If $a$ and $b$ are two solutions of $\,e^x \cos x -1=0$, then how many solutions of the equation $e^x \sin x-1=0$ lie between $a$ and $b$ ?
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### A Ferris wheel has a radius of 20 m. Passengers get on halfway up on the right side.

A Ferris wheel has a radius of 20 m. Passengers get on halfway up on the right side. The direction of rotation is counter clockwise. The bottom of the Ferris wheel is 2 m off the ground. It rotates ...
221 views

### Proving $f(x) := \sin(ax) - \sin(bx)$ has absolute value greater or equal to 1 for some $x>0$

I am trying to prove that for any choice of reals $a$ and $b$ such that $a \neq b$ the function $f(x) := \sin(ax) - \sin(bx)$ has $|f(x)| \geq 1$ for some $x>0$ (This is not an exercise question ...
874 views

### period of a function $\lvert\sin x\rvert+\lvert\cos x\rvert$

I have read that $$y=\lvert\sin x\rvert+ \lvert\cos x\rvert$$ is periodic with fundamental period $\frac{\pi}{2}$. But Wolfram says it is periodic with period $\pi$. Please tell what is correct.
Let $z=e^{it}+1$ where $0\leq t\leq \pi$, Find the trigonometric representation of $z^2+z+1$. (The trigonometric representation should be in the form of : $r(\cos \theta +i \sin \theta)$, where ...