GENERAL SOLUTION OF TRIGNOMETRIC

EQUATIONSEQUATIONS

S l i ( )Solution (a)

, then x isIf , then x is If

(a) 2nπ (b) nπ

(c) (2n+1)π (d) ( ) ( ) ( )

Put n = 0Put n = 0,

Solution (d)

(a) (b)

(c) (d)

Solution (a)

SOLUTION (a)SOLUTION (a)

( ) l(a) One real root 

(b) Two real root(b) Two real root 

(c) More than one real root 

(d) No real root

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 z1

and 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 z1

and 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, -1≤y≤1

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

Solution (b)

(a) 0 (b) 2( ) ( )

(c) 1 (d) 1(c) 1 (d) – 1

If α1, α2, … , αn are the nth

roots of unity then

(1+ α1 )(1+ α2) … (1+ αn) =

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|= 0 if and only if z = 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 argumentProperties of argument 

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

Arg(zn) = n arg(z)

Solution (a)Solution (a)

Solution(c)

Solution (c)Solution (c)