Prove the formulae: \(\arcsin (- x) = - \arcsin x\), \(\arccos (- x) = \pi - \arccos x\).
In the centre of a rectangular billiard table that is 3 m long and 1 m wide, there is a billiard ball. It is hit by a cue in a random direction. After the impact the ball stops passing exactly 2 m. Find the expected number of reflections from the sides of the table.
The number \(x\) is such that both the sums \(S = \sin 64x + \sin 65x\) and \(C = \cos 64x + \cos 65x\) are rational numbers.
Prove that in both of these sums, both terms are rational.
Find the locus of points whose coordinates \((x, y)\) satisfy the relation \(\sin(x + y) = 0\).
Does there exist a flat quadrilateral in which the tangents of all interior angles are equal?