GENERAL SOLUTION OF TRIGNOMETRIC .Geometrical interpretation If z z are two complex Geometrical...

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Transcript of GENERAL SOLUTION OF TRIGNOMETRIC .Geometrical interpretation If z z are two complex Geometrical...

  • GENERAL SOLUTION OF TRIGNOMETRIC

    EQUATIONSEQUATIONS

  • S l i ( )Solution (a)

  • , then x isIf , then x is If

    (a) 2n (b) n

    (c) (2n+1) (d) ( ) ( ) ( )

  • Put n = 0Put n = 0,

    Solution (d)

  • (a) (b)

    (c) (d)

  • Solution (a)

  • SOLUTION (a)SOLUTION (a)

  • ( ) l(a)Onerealroot

    (b) Two real root(b)Tworealroot

    (c)Morethanonerealroot

    (d)Norealroot

  • Solution (d)

  • Solution (b)

  • Solution (a)

  • Solution (c)Solution (c)

  • Complex numbers

  • Solution (a)

  • Solution (b)So ut o (b)

  • Solution (a)

  • Solution (b)Solution (b)

  • Solution (c)Solution (c)

  • Solution (c)Solution (c)

  • Solution (c)Solution (c)

  • Solution (c) ( )

  • Solution (b)

  • Solution(b)

  • Solution (c)Solution (c)

  • G t i l i t t tiGeometrical interpretationof complex numbersp

    If z1 ,z2 are two complex numbers, 1 2 the locus of point P(z) such that |z-z1|+|z - z2|=2a where 2a > |z1 z2| represents an ellipse with foci zrepresents an ellipse with foci z1and z2 .2

  • Geometrical interpretation

    If z z are two complex

    Geometrical interpretationof complex numbers

    If z1 ,z2 are two complex numbers , the locus of point P( ) h h | | | | 2P(z) such that |z - z1|-|z - z2|=2a where 2a < |z1 z2| represents 1 2

    an hyperbola with foci z1 and z2 . If |z-z1|=k |z-z2| represents aIf |z-z1|=k |z-z2| represents a

    circle if k 1 and a straightline if k =1line if k =1

  • Geometrical interpretationGeometrical interpretationof complex numbers

    |z-z1|2 + |z-z2|2 =|z1 z2|2

    represents a circle withrepresents a circle withdiametric ends z1 and z2 .

    |z-z1|= |z-z2| represents the perpendicular bisector of z1and z2and z2 .

  • Geometrical interpretationGeometrical interpretationof complex numbers

    locus of z satisfying represents a circle withrepresents a circle withz1 and z2 as ends of diameter.

    Locus of z satisfying represents a straight line whichpasses through z1 and z2 .p g 1 2

  • 5

    5 (3,4)(3,4)

    Solution (c)

  • Solution (a)

  • Solution (a)

  • Solution (a)Solution (a)

  • Solution (b)

  • Solution (b)( )

  • (a) 0 (b) 90

    (c) 180 (d) - 90(c) 180 (d) 90

  • S l ti ( )Solution (c)

  • (a) 4 (b) 2

    (c) 1 (d) 8(c) 1 (d) 8

  • Solution (c)( )

  • (a) w or w2

    (b) w or w2

    (c) 1+i or 1 i(c) 1+i or 1- i(d) 1 + i or 1 i( )

  • Solution (a)( )

    *

  • (a) Re(z)=1, Im(z) = 2 (b) Re(z)=1, -1y1

    (c) Re(z)+Im(z) = 0 (d) none

  • Solution (b)

  • (a) 0 (b) 2( ) ( )

    (c) 1 (d) 1(c) 1 (d) 1

  • If 1, 2, , n are the nth

    roots of unity then

    (1+ 1 )(1+ 2) (1+ n) =

    0 if n is even and 2 if n is odd

    Solution (a)( )

  • Solution (a)Solution (a)

  • Solution (a)Solution (a)

  • Solution (a)Solution (a)

  • Solution (c)Solution (c)

  • Solution (d)Solution (d)

  • Properties of modulusProperties of modulus|z|=0ifandonlyifz=0|z|=||=|z|=|||z1 z2 z3 .zn|=|z1||z2||z3||zn|| 1 2 3 n| | 1|| 2|| 3| | n||zn|=|z|n

    |z1 + z2 + z3 + + z | |z1|+|z2|+|z3|+ +|z ||z1 +z2 +z3 ++zn ||z1|+|z2|+|z3|++|zn||z1 +z2|||z1||z2||

  • Properties of argumentPropertiesofargument

    Arg(z1z2z3 z ) = arg(z1)arg(z2)arg(z3) arg(z )Arg(z1z2z3 .zn)=arg(z1)arg(z2)arg(z3)arg(zn)

    Arg(zn)=narg(z)

  • Solution (a)Solution (a)

  • Solution(c)

  • Solution (c)Solution (c)