ComplexNumber Trigonometry AMC10/12

2015

Problem - 376

Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\frac{n}{d}$ where $n$ and $d$ are integers with $1 \le d \le 5$. What is the probability that \[(\text{cos}(a\pi)+i\text{sin}(b\pi))^4\] is a real number?