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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)