Intereting Posts

Liouville's theorem for subharmonic functions
Proving an Combination formula $ \binom{n}{k} = \binom{n-1}{k}+\binom{n-1}{k-1}$
Convergence in $L^p$ and convergence almost everywhere
Changing order of integration (multiple integral)
existence of solution of a degenerate pde with change of variables
Basic question on primitive roots
Finer topologies on a compact Hausdorff space
The definition of span
Example that a measurable function $f$ on $[1,\infty )$ can be integrable when $\sum _{n=1}^{\infty }\int_{n}^{n+1}f$ diverges.
Applications for Homology
Prove that $f(x)=d(x,A)=\inf_{y\in A}d(x,y)$ is continuous on $M$
Finding the inverse of $f(x) = x^3 + x$
Showing that $\sum_{i=1}^n \frac{1}{i} \geq \log{n}$
Dimension inequality for homomorphisms between noetherian local rings
Prove that Baire space $\omega^\omega$ is completely metrizable?

In showing that the trisection of an angle with ruler and compass is not possible in general one shows that $\cos(20^\circ)$ cannot be constructed (thus the angle $60^\circ$ cannot be trisected) by determining its minimal polynomial, which is $8x^3-6x-1$.

Solving $8x^3-6x-1=0$ yields a solution $x_1=\sqrt[3]{\frac{1}{16}+\frac{\sqrt{3}}{16}i}+\sqrt[3]{\frac{1}{16}-\frac{\sqrt{3}}{16}i}$. Expressing $\sqrt[3]{\frac{1}{16}+\frac{\sqrt{3}}{16}i}$ and $\sqrt[3]{\frac{1}{16}-\frac{\sqrt{3}}{16}i}$ in polar form yields $x_1=\cos(20^\circ)$.

Is it possible to express $\cos(20^\circ)$ with radicals without complex numbers?

- Is there a name for this strange solution to a quadratic equation involving a square root?
- Show that $\lim_{n\rightarrow \infty} \sqrt{c_1^n+c_2^n+\ldots+c_m^n} = \max\{c_1,c_2,\ldots,c_m\}$
- Limit of the sequence $\lim_{n\rightarrow\infty}\sqrtn$
- Is $n^{th}$ root of $2$ an irrational number?
- How do I solve this System of Equations?
- Does this equality hold: $\frac{\sqrt{2|x|-x^2}-0}x = \sqrt{\pm\frac2x-1}$?

- Help finding solution for trigonometric equation
- Simplifying Trig Product in terms of a single expression and $n$
- Find a formula for a sequence $\{\sqrt{3},\sqrt{3\sqrt{3}},\sqrt{3\sqrt{3\sqrt{3}}},…\}$
- Derive $\frac{d}{dx} \left = \frac{1}{\sqrt{1-x^2}}$
- How to evaluate the limit $\lim_{x\to 0} \frac{1-\cos(4x)}{\sin^2(7x)}$
- How to prove $x=120^\circ$
- Solve the equation: $\cos^2(x)+\cos^2(2x)+\cos^2(3x)=1$
- Evaluating $\int_0^{\large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $
- Positivness of the sum of $\frac{\sin(2k-1)x}{2k-1}$.
- Solve this tough fifth degree equation.

$$

\cos (20^\circ) = \cos (\pi/9) = -\frac12 (-1)^{8/9} \left(1+(-1)^{2/9}\right)

$$

- Are there spaces “smaller” than $c_0$ whose dual is $\ell^1$?
- Why does $a^{b}·a^{c}=a^{b+c}$?
- Namesake of Cantor's diagonal argument
- How can I show that $ab \sim \gcd (a,b) {\operatorname{lcm} (a,b)}$ for any $a,b \in R \setminus \{0\}$?
- Combinatorial Proof of Multinomial Theorem – Without Induction or Binomial Theorem
- How to prove that there exist infinitely many integer solutions to the equation $x^2-ny^2=1$ without Algebraic Number Theory
- Significance of Sobolev spaces for numerical analysis & PDEs?
- Could a square be a perfect number?
- when is $\frac{1}{n}\binom{n}{r}$ an integer
- Are there vector bundles that are not locally trivial?
- Fourier transform of sine and cosine function
- Why hyperreal numbers are built so complicatedly?
- Eventually constant variable assignments
- Continued fraction of a square root
- Solving $ax \equiv c \pmod b$ efficiently when $a,b$ are not coprime