Complex Analysis- Lecture 21 - Trigonometric Function

3.3 Trigonometric Function:-
From Euler formula, we get
e^ix=cos?x+i sin?x
e^(-ix)=cos?x-i sin?x
Thus sin?x=(e^ix-e^(-ix))/2i,cos??x=? (e^ix+e^(-ix))/2
If we suppose Z=x+iy , we get
sin?Z=(e^iZ- e^(-iZ))/2i,cos??Z=? (e^iZ+e^(-iZ))/2
3.3.1 Characteristics:-
tan??Z=sin?Z/cos??Z ? ? ,cos??Z?0?
? cot??Z=cos?Z/sin?Z ,sin??Z?0?
sec?Z=1/cos?Z ,cos?Z?0
csc?Z=1/sin?Z ,sin?Z?0
tan?Z ,sec?Z are analytic function in all point Zplan except? cos??Z=0.
Also ? cot??Z,csc?Z are analytic functions in all point Zplans? sin???Z=0?.
Differential
d/dZ (sin?Z )=cos?Z , d/dZ (cos?Z )=-sin?Z
d/dZ (tan?Z )=sec^2?Z , d/dZ (cot?Z )=-csc^2?Z
d/dZ (sec?Z )=sec?Z tan?Z ,d/dZ (csc?Z )=- csc?Z cot?Z
Relations
sin^2?Z+cos^2?Z=1
sin?(-Z)=-sin?(Z)
cos?(-Z)=cos?Z
sin?(Z_1±Z_2 )=sin??Z_1 ? cos??Z_2 ?±sin??Z_2 ? cos??Z_1 ?
cos?(Z_1±Z_2 )=cos??Z_1 cos??Z_2 ? ??sin??Z_1 sin??Z_2 ? ?
cos?(2Z)=cos^2?Z-sin^2?Z
sin?(2Z)=2sin?Z cos?Z
How make sin?Z=u+iv
sin?Z=1/2i (e^iZ-e^(-iZ) )=1/2i (e^i(x+iy) -e^(-i(x+iy) ) )
=1/2i (e^(-y+ix)-e^(y-ix) )=1/2i (e^(-y) [cos?x+i sin??x]?-e^y [cos?x-i sin??x]? )
=1/2i (e^(-y) cos??x+i? e^(-y) sin?x-e^y cos??x+i? ?e^y sin??x )
=1/2i (cos?x (e^(-y)-e^y )+i sin?x (e^(-y)-e^y ) )
=1/i cos??x ((e^(-y)-e^y ))/2+? sin?x ((e^(-y)+e^y ))/2
=-i cos??x ((e^(-y)-e^y ))/2+? sin?x cosh?y
=sin?x cosh?y+i cos??x ((?e^y-e?^(-y) ))/2 ?
?(sin?Z=sin?x ? cosh??y+i cos??x ? sinh??y ? )
H.W:- cos?z=cos?x cosh?y-i sin?x sinh?y
H.W:- |sin?Z |,|cos?Z |
EX:-
|sin?Z |^2=(sin?x cosh?y )^2+(cos?x sinh?y )^2
=sin^2?x cosh^2?y+cos^2?x sinh^2?y
=sin^2?x (1+sinh^2?y )+(1-sin^2?x ) sinh^2?y
=sin^2?x+sin^2??x sinh^2?y+? sinh^2?y-sin^2?x sinh^2?y
=sin^2?x+sinh^2?y
|cos?Z |^2=cos^2?x+sinh^2?y
EX (1):- Prove that
(sin?Z ) ?=sin?Z ?
(sin?Z ) ?=(sin?x cosh?y+i cos?x sinh?y ) ?=sin??x cosh?y-i cos??x sinh?y…*? ?
sin?Z ? =sin??(x-iy)=sin?(x+i(-y) ) ?
=sin?x cosh?(-y)+i cos?x sinh?(-y)
=sin?x cosh?y-i cos?x(-sinh?y )
=sin?x cosh?y-i cos??x sinh?y ?………**
(cos?Z ) ?=cos?Z ? H.W
Ex (2):- Write the following function as u+iv ?
cos?i
x=0 & y=1
cos?Z=cos?x cosh?y-i sin?x sinh?y
cos?i = cos?0 cosh?1-i sin?0 sinh?1
=1 cosh?1-i(0) sinh?1=cosh?1
sinh?(?i) H.W
sec?(1-i) H.W
Since
sin?Z=sin?x cosh?y+i cos?x sinh?y
cos?Z=cos?x cosh??y-i? sin??x sinh?y ?
If x=0 , then
sin?iy=i sinh?y
cos?iy=cosh?y Put t=iy , then
sin??t=-i? sinh?it
cos?t=cosh?? it?