Does there exist a function \(f (x)\) defined for all \(x \in \mathbb{R}\) and for all \(x, y \in \mathbb{R}\) satisfying the inequality \(| f (x + y) + \sin x + \sin y | < 2\)?
Does there exist a real number \({\alpha}\) such that the number \(\cos {\alpha}\) is irrational, and all the numbers \(\cos 2{\alpha}\), \(\cos 3{\alpha}\), \(\cos 4{\alpha}\), \(\cos 5{\alpha}\) are rational?
Calculate \(\int_0^{\pi/2} (\sin^2 (\sin x) + \cos^2 (\cos x))\,dx\).
Prove the irrationality of the following numbers:
a) \(\sqrt{3}{17}\)
b) \(\sqrt{2} + \sqrt{3}\)
c) \(\sqrt{2} + \sqrt{3} + \sqrt{5}\)
d) \(\sqrt{3}{3} - \sqrt{2}\)
e) \(\cos 10^{\circ}\)
f) \(\tan 10^{\circ}\)
g) \(\sin 1^{\circ}\)
h) \(\log_{2}3\)
Prove that for \(x \ne \pi n\) (\(n\) is an integer) \(\sin x\) and \(\cos x\) are rational if and only if the number \(\tan x/2\) is rational.
A square grid on the plane and a triangle with vertices at the nodes of the grid are given. Prove that the tangent of any angle in the triangle is a rational number.
Find the largest and smallest values of the functions
a) \(f_1 (x) = a \cos x + b \sin x\); b) \(f_2 (x) = a \cos^2x + b \cos x \sin x + c \sin^2x\).
Prove that the function \(\cos \sqrt {x}\) is not periodic.
Prove the formulae: \(\arcsin (- x) = - \arcsin x\), \(\arccos (- x) = \pi - \arccos x\).
Solve the system of equations: \[\begin{aligned} \sin y - \sin x &= x-y; &&\text{and}\\ \sin y - \sin z &= z-y; && \text{and}\\ x-y+z &= \pi. \end{aligned}\]