# Interesting Concepts

1. Dec 21, 2009

### FeDeX_LaTeX

Hello, I have a few questions that I have made myself, but I don't know how to answer them.

1) Say that there is an isosceles triangle ABC. Two of the equal angles are 1/∞. What is the other angle?

2) Is a sphere really a sphere if you can draw a tangent to it?

3) Assume that there exists a perfect sphere, that can have no tangent drawn to it. If one were to drop the sphere, what would happen to it? It couldn't touch the ground, because it can't have a tangent drawn to it.

4) The graph of tan(x) has several vertical asymptotes on it. But what is the width of this asymptote? Moreover, what is the distance between the edge of the curve on the left-hand side of the asymptote and the edge of the curve on the right-hand side (at the centre)?

5) $$F_{n}=\frac{\varphi^n - (1-\varphi)^n}{\sqrt{5}}$$

Shown above is the formula for the Fibonacci sequence. But how is this derived?

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I'll have more questions soon, but these ideas have been boggling my mind for a while now. Can anyone help me out?

2. Dec 21, 2009

### fourier jr

Since the sum of angles of a triangle is 180 degrees, or $\pi$ radians, I would say the answer is $\pi$, or rather, $\lim_{x \rightarrow \infty} (\pi - 2/x) = \pi$ while the sum of the other angles, $\lim_{x \rightarrow \infty} (1/x + 1/x)$ goes to 0

Why do you think a sphere can't have tangent lines? At a given point, a point on the surface of a sphere has infinitely many tangent lines, which make up the tangent plane at that point

an asymptote is a line, so it has the width of a line, which I guess is 0. since tan(x) has asymptotes at the points $\frac{(2k+1)\pi}{2}$, the 2nd part sounds like you're asking about the distance between the values $$\tan \left(\frac{(2k+1)\pi}{2} \pm \ell \right)$$ for $|\ell| < \pi$