Just wanting to clarify how an exponential representation can be simplified. For example in a question regarding Fourier coefficients the form of the coefficients are $X_{-1} = -2.5e^{-j\pi/6}$ and $X_1 = -2.5e^{j\pi/6}$. But in the solutions they are simplified to $X_{-1} = 2.5e^{5j\pi/6}$ and $X_1 = 2.5e^{-5j\pi/6}$.
So from what i can see, for the $-1$ in front of the exponential, it was recognised as $-1=e^{j\pi}$, so there was an addition of $\pi$ to each. But for $X_1$ shouldn't that mean it becomes $2.5e^{7\pi j/6}$?
I'm thinking the authors wanted to show the angles as those in opposite phase. If you take a look at the angles, $-\pi/6$ is coterminal with $5 \pi/6$, while $\pi/6$ is coterminal with $-5\pi/6$.
While $j7 \pi/6$ is correct, in order for the answer to show opposite phases, the authors used $-j5 \pi /6$ instead.