A trigonometric value

Evaluate

$$\mathcal{V}= \tan \left( 2 \arcsin \frac{1}{\sqrt{5}}\right)$$

Solution

Trigonometric sum

Evaluate the value of

$$\mathcal{A}=\cos \frac{2\pi}{13} + \cos \frac{6 \pi}{13} + \cos \frac{8\pi}{13}$$

Solution

A determinant on a triangle

Let a triangle have angles $A, B, C$. Prove that:

$$\begin{vmatrix}
1 &\sin A  & \cot \frac{A}{2} \\\\
 1& \sin B & \cot \frac{B}{2}\\\\
 1& \sin C & \cot \frac{C}{2}
\end{vmatrix}=0$$

Solution

Rational points

How many rational points do there exist on the sphere

$$x^2+y^2+z^2=7$$

Solution

Rational number

Let $0, a_1a_2a_3\dots$ the decimal expansion of a number. Can you suggest such a way that when the order of the digits is changed the new number is rational ?

Solution

Integer part of a number

Let $\left \lfloor \cdot \right \rfloor$ denote the floor function. Evaluate the integer part of the number:

$$\left \lfloor \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}}+ \cdots + \frac{1}{\sqrt{2010}} \right \rfloor$$

Solution

Transcedental roots

Prove that the non zero roots of the equation $\tan x =x$ are transcedental.

Solution

Equality of inverse trig. functions

Prove that:

$$\arccos \left ( \frac{1}{5} \right )= 2\arctan  \left ( \sqrt{\frac{2}{3}} \right )$$

Solution