Calculating the magnitude spectrum and phase spectrum
HINT:

Magnitude $M(\omega)\ge 0$ and phase $\phi(\omega)$ are defined by

$$X(\omega)=M(\omega)e^{j\phi(\omega)}\tag{1}$$

Note that $(1)$ is generally complex-valued. In your example, the Fourier transform $X(\omega)$ is clearly real-valued. This restricts the possible values of the phase $\phi(\omega)$. What are those two values of $\phi(\omega)$ for which $X(\omega)$ in $(1)$ is real-valued? How do they relate to the sign of $X(\omega)$? I'm sure you can take it from here.