The period of F(x+a)=-f(x) is 2a. Proof process: because f(x+a)=-f(x) and f(x)=-f(x-a), so f(x+a)=f(x-a), that is, f(x+2a)=f(x), so the period is 2a.

The function period formula of sinx is T=2π, and sinx is a sine function with a period of 2π.

The function period formula of cosx is T=2π, and cosx is a cosine function with a period of 2π.

The function period formula of tanx and cotx is T=π, and tanx and cotx are tangent and cotangent respectively.

The function period formula of secx and cscx is T=2π, and secx and cscx are secant cotangent.

Extended data:

Y=Asin(wx+b) Periodic formula T=2π/w

Y=Acos(wx+b) periodic formula T=2π/w

Y=Atan(wx+b) periodic formula T=π/w

Important inference:

If the function f(x)(x∈D) has two symmetry axes x=a and x=b in the definition domain, then the function f(x) is a periodic function with a period T=2|b-a| (not necessarily the minimum positive period).

If the function f(x)(x∈D) has two symmetrical centers A(a, 0) and B(b, 0) in the defined domain, then the function f(x) is a periodic function with a period T=2|b-a| (not necessarily the minimum positive period).

If the function f(x)(x∈D) has an axis of symmetry x=a and a center of symmetry B(b, 0)(a≠b) in the defined domain, then the function f(x) is a periodic function with a period T=4|b-a| (not necessarily the minimum positive period).