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Math · Advanced Math
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15. Define the functions (2k) 24+1 (i) Prove that C and S are indefinitely differentiable on IR (ii) Prove that, for, all .r e lR we have; Cr1i(r) =-S(x), C(2)(r)--C(r), C(3)(z) (i) Prove that, for any ay aE R we have (iv) Show that C is even and S is odd (v) Show that for all r ER we have: (vi) Show that there exists a unique number ξ e (0,2) such that C() 0 (vii) Let π :-: 2, where ξ is the number obtained at (vi). Show that S(π/2)-1, C(r)=-1. S(r) = 0,C(2n) = 1, and S(2t)-0. (vii) Show that or all r e R we have S), C) C(T-)-Cr), S( S(x), E(z + 2r)-C(z), S(z + 2π)-S(2). (ix) Let r, y be two real numbers such that r2 + y 1. Show that, there exists a unique number t E [0,2T) such that r - C(t) and y St).

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