Upload
trinhminh
View
221
Download
2
Embed Size (px)
Citation preview
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)