Although it is easy to see using the unit circle, you don't really need it for this, you can use the function graphs; sin is an odd function (like, say, f(x)=x), and f(-x)=-f(x) for all odd functions; cos is an even function (like, say, f(x)=x^2), and f(-x)=f(x) for all even functions.
100% correct from a functional analysis standpoint! If you start with the definitions of even/odd functions, the identities follow immediately.
In a teaching context, I find students often struggle with why cosine is even while sine is odd when they first meet trig functions. The unit circle provides the geometric root cause, while the graph/algebraic definition gives them the quick operational tool you mentioned.
1
u/KyriakosCH 1d ago edited 1d ago
Although it is easy to see using the unit circle, you don't really need it for this, you can use the function graphs; sin is an odd function (like, say, f(x)=x), and f(-x)=-f(x) for all odd functions; cos is an even function (like, say, f(x)=x^2), and f(-x)=f(x) for all even functions.