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

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GENERAL SOLUTION OF TRIGNOMETRIC EQUATIONS EQUATIONS

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

Page 1: GENERAL SOLUTION OF TRIGNOMETRIC EQUATIONS · Geometrical interpretation If z z are two complex Geometrical interpretation of complex numbers If z1,z2 are two complex numbers , the

GENERAL SOLUTION OF TRIGNOMETRIC

EQUATIONSEQUATIONS

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S l i ( )Solution (a)

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, then x isIf , then x is If

(a) 2nπ (b) nπ

(c) (2n+1)π (d) ( ) ( ) ( )

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Put n = 0Put n = 0,

Solution (d)

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(a) (b)

(c) (d)

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Solution (a)

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SOLUTION (a)SOLUTION (a)

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( ) l(a) One real root 

(b) Two real root(b) Two real root 

(c) More than one real root 

(d) No real root

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Solution (d)

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Solution (b)

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Solution (a)

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Solution (c)Solution (c)

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Complex numbers

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Solution (a)

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Solution (b)So ut o (b)

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Solution (a)

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Solution (b)Solution (b)

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Solution (c)Solution (c)

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Solution (c)Solution (c)

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Solution (c)Solution (c)

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Solution (c) ( )

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Solution (b)

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Solution(b)

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Solution (c)Solution (c)

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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

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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

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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 .

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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

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5

5 (3,4)(3,4)

Solution (c)

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Solution (a)

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Solution (a)

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Solution (a)Solution (a)

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Solution (b)

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Solution (b)( )

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(a) 0 (b) 90°

(c) 180° (d) - 90°(c) 180 (d) 90

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S l ti ( )Solution (c)

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(a) 4 (b) 2

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

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Solution (c)( )

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(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( )

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Solution (a)( )

*

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(a) Re(z)=1, Im(z) = 2 (b) Re(z)=1, -1≤y≤1

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

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Solution (b)

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

(c) 1 (d) 1(c) 1 (d) – 1

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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)( )

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Solution (a)Solution (a)

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Solution (a)Solution (a)

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Solution (a)Solution (a)

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Solution (c)Solution (c)

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Solution (d)Solution (d)

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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||

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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)

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Solution (a)Solution (a)

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Solution(c)

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Solution (c)Solution (c)