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 $a+b?$

Give your answer to 3 decimal places.