Let the number \(\alpha\) be given by the decimal:
a) \(0.101001000100001000001 \dots\);
b) \(0.123456789101112131415 \dots\).
Will this number be rational?
For what natural numbers \(a\) and \(b\) is the number \(\log_{a} b\) rational?
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.
Derive from the theorem in question 61013 that \(\sqrt{17}\) is an irrational number.
The numbers \(x\), \(y\) and \(z\) are such that all three numbers \(x + yz\), \(y + zx\) and \(z + xy\) are rational, and \(x^2 + y^2 = 1\). Prove that the number \(xyz^2\) is also rational.
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.
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?
Are there any irrational numbers \(a\) and \(b\) such that the degree of \(a^b\) is a rational number?