Question about Unit Circle and Trig

Question

I'm wondering about the difference between $-45 ^{\circ}$ and $315 ^{\circ}$. As I'm reading about it and it doesn't make sense to me if they both land at the same place, there shouldn't be a difference.

But if there isn't a difference then $\frac{7\pi}{4}$ should equal $\frac{-\pi}{4}$ which isn't the case right? Would really appreciate some clearing up here. Thanks in advance.

Answer 1

I the x-y plane both angles describe the same ray line but when we consider the angle as the variable of a function then $-\frac{\pi}{4}$ and $\frac{7\pi}{4}$ represent two distinct values.

More in general for any angle $\theta$ we have that

• in x-y plane all angles $\theta+2\pi k$ represents the same ray line
• as variable of a function $f(\theta)$, angles $\theta+2\pi k$ represent distinct values