x2 t01 04 mod arg form(2013)
TRANSCRIPT
Mod-Arg Form
Mod-Arg Form
y
xO
ModulusThe modulus of a complex number is the length of the vector OZz = x + iy
x
y
Mod-Arg Form
y
xO
ModulusThe modulus of a complex number is the length of the vector OZz = x + iy
x
y 22
222
yxr
yxr
zr
Mod-Arg Form
y
xO
ModulusThe modulus of a complex number is the length of the vector OZz = x + iy
x
y 22
222
yxr
yxr
zr
22 yxz
Mod-Arg Form
y
xO
ModulusThe modulus of a complex number is the length of the vector OZz = x + iy
x
y 22
222
yxr
yxr
zr
22 yxz
ArgumentThe argument of a complex number is the angle the vector OZ makes with the positive real (x) axis
zarg
Mod-Arg Form
y
xO
ModulusThe modulus of a complex number is the length of the vector OZz = x + iy
x
y 22
222
yxr
yxr
zr
22 yxz
ArgumentThe argument of a complex number is the angle the vector OZ makes with the positive real (x) axis
zarg
xyz 1tanarg zarg
ige 44ofargument andmodulus theFind ..
22 4444 i
ige 44ofargument andmodulus theFind ..
2432
22 4444 i
ige 44ofargument andmodulus theFind ..
44tan44arg 1i
2432
1tan 1
22 4444 i
ige 44ofargument andmodulus theFind ..
44tan44arg 1i
2432
1tan 1
4
22 4444 i
ige 44ofargument andmodulus theFind ..
44tan44arg 1i
2432
1tan 1
4
Every complex number can be written in terms of its modulus and argument
22 4444 i
iyxz
ige 44ofargument andmodulus theFind ..
44tan44arg 1i
2432
1tan 1
4
Every complex number can be written in terms of its modulus and argument
sincos irr sincos ir
22 4444 i
iyxz
zzrarg
where;
ige 44ofargument andmodulus theFind ..
44tan44arg 1i
2432
1tan 1
4
Every complex number can be written in terms of its modulus and argument
sincos irr sincos ir
The mod-arg form of z is; sincos irz
rcisz
iie.g. 44
iie.g. 44
424 cis
iie.g. 44
424 cis
iii 3
iie.g. 44
424 cis
iii 3
24
13 22
z
iie.g. 44
424 cis
iii 3
24
13 22
z3
1tanarg 1z
6
iie.g. 44
424 cis
iii 3
24
13 22
z3
1tanarg 1z
6
623 cisi
iie.g. 44
424 cis
iii 3
24
13 22
z3
1tanarg 1z
6
623 cisi
Convert 6cis to Cartsian form 6
ii
iie.g. 44
424 cis
iii 3
24
13 22
z3
1tanarg 1z
6
623 cisi
Convert 6cis to Cartsian form 6
ii
6cis 6(cos sin )6 6 6
i
iie.g. 44
424 cis
iii 3
24
13 22
z3
1tanarg 1z
6
623 cisi
Convert 6cis to Cartsian form 6
ii
6cis 6(cos sin )6 6 6
i
3 16( )2 2
3 3 3
i
i
Mod-Arg Relations
Mod-Arg Relations 2121 1 zzzz
2121 argargarg zzzz
Mod-Arg Relations 2121 1 zzzz
2121 argargarg zzzz
NOTE: Multiplication rotates z1 by arg z2
Mod-Arg Relations 2121 1 zzzz
222111 andlet cisrzcisrz
2121 argargarg zzzz
22211121 sincossincos irirzz Proof:
NOTE: Multiplication rotates z1 by arg z2
Mod-Arg Relations 2121 1 zzzz
222111 andlet cisrzcisrz
2121 argargarg zzzz
22211121 sincossincos irirzz Proof:
2121212121 sinsinsincoscossincoscos iirr 2121212121 sincoscossinsinsincoscos irr
NOTE: Multiplication rotates z1 by arg z2
Mod-Arg Relations 2121 1 zzzz
222111 andlet cisrzcisrz
2121 argargarg zzzz
22211121 sincossincos irirzz Proof:
2121212121 sinsinsincoscossincoscos iirr 2121212121 sincoscossinsinsincoscos irr
212121 sincos irr
NOTE: Multiplication rotates z1 by arg z2
Mod-Arg Relations 2121 1 zzzz
222111 andlet cisrzcisrz
2121 argargarg zzzz
22211121 sincossincos irirzz
21
2121
zzrrzz
Proof:
2121212121 sinsinsincoscossincoscos iirr 2121212121 sincoscossinsinsincoscos irr
212121 sincos irr
NOTE: Multiplication rotates z1 by arg z2
Mod-Arg Relations 2121 1 zzzz
222111 andlet cisrzcisrz
2121 argargarg zzzz
22211121 sincossincos irirzz
21
2121
zzrrzz
Proof:
2121212121 sinsinsincoscossincoscos iirr 2121212121 sincoscossinsinsincoscos irr
212121 sincos irr
21
2121
argarg arg
zzzz
NOTE: Multiplication rotates z1 by arg z2
Mod-Arg Relations 2121 1 zzzz
222111 andlet cisrzcisrz
2121 argargarg zzzz
22211121 sincossincos irirzz
21
2121
zzrrzz
Proof:
2121212121 sinsinsincoscossincoscos iirr 2121212121 sincoscossinsinsincoscos irr
212121 sincos irr
21
2121
argarg arg
zzzz
NOTE: it follows that;nn zzzzzzzz 321321
nn zzzzzzzz argargargargarg 321321
NOTE: Multiplication rotates z1 by arg z2
2
1
2
1 2zz
zz
212
1 argargarg zzzz
2
1
2
1 2zz
zz
212
1 argargarg zzzz
NOTE: it follows that;
43
21
43
21
zzzz
zzzz
432143
21 argargargargarg zzzzzzzz
2
1
2
1 2zz
zz
212
1 argargarg zzzz
NOTE: it follows that;
43
21
43
21
zzzz
zzzz
432143
21 argargargargarg zzzzzzzz
nn zz 3
znzn argarg
i
iize.g.2325 ofargument and modulus theFind
i
iize.g.2325 ofargument and modulus theFind
22
2222
231215
z
i
iize.g.2325 ofargument and modulus theFind
22
2222
231215
z
13526
10
i
iize.g.2325 ofargument and modulus theFind
22
2222
231215
z
13526
10
32tan
21tan
51tanarg 111z
i
iize.g.2325 ofargument and modulus theFind
22
2222
231215
z
13526
10
32tan
21tan
51tanarg 111z
84175
1433621539111
i
iize.g.2325 ofargument and modulus theFind
22
2222
231215
z
13526
10
32tan
21tan
51tanarg 111z
84175
1433621539111
Patel: Exercise 4B; evens
Patel: Exercise 4C; 1 to 10 evens
Cambridge: Exercise 1D; 9, 11, 13, 14, 16, 20