Root of unity problem for wiki

Algebra Level 2

Let \(z\) be a \(7^\text{th}\) root of unity. In other words, \(z\) is a complex number that satisfies the equation \(z^7 = 1\).

If \(z=a+bi\), where \(a\) and \(b\) are real numbers, then what is the maximum value of \(a+b\)?