What is the degree of the circle function $A: S^1 \rightarrow S^1$, $A(\theta)=\theta+\pi$?
Considering coefficient of $\theta$ to be $1$ and ignoring translation $\pi$ so $deg (A)=1$. On the other hand, $A$ is the antipodal map that maps each point on the circle to the point opposite it through the center so is like $f(\theta)=-\theta$ so should be $deg(A)=-1$?
The degree of a map is a homotopy invariant, and since your map $A$ is homotopic to the identity (can you see why?) the degree should be $1$.
The degree of the antipodal map is not always $-1$. It depends on the dimension of the sphere on which it is defined. On the $1$-sphere, the circle, it has degree $1$. Can you find a formula for the degree of the antipodal map on the $n$-sphere $S^n$?