x2 t01 11 nth roots of unity (2012)
TRANSCRIPT
nth Roots Of Unity 1nz
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
When placed on an Argand Diagram, they form a regular n sided polygon, with vertices on the unit circle.
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
1 .. 5 zge
When placed on an Argand Diagram, they form a regular n sided polygon, with vertices on the unit circle.
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
1 .. 5 zge
When placed on an Argand Diagram, they form a regular n sided polygon, with vertices on the unit circle.
2 0 0, 1, 25kz cis k
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
1 .. 5 zge
When placed on an Argand Diagram, they form a regular n sided polygon, with vertices on the unit circle.
2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5
z cis cis cis cis cis
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
1 .. 5 zge
When placed on an Argand Diagram, they form a regular n sided polygon, with vertices on the unit circle.
y
x0cis
2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5
z cis cis cis cis cis
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
1 .. 5 zge
When placed on an Argand Diagram, they form a regular n sided polygon, with vertices on the unit circle.
y
x0cis
52cis
52
cis
2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5
z cis cis cis cis cis
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
1 .. 5 zge
When placed on an Argand Diagram, they form a regular n sided polygon, with vertices on the unit circle.
y
x0cis
52cis
52
cis
54cis
54
cis
2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5
z cis cis cis cis cis
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
1 .. 5 zge
When placed on an Argand Diagram, they form a regular n sided polygon, with vertices on the unit circle.
y
x0cis
52cis
52
cis
54cis
54
cis
2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5
z cis cis cis cis cis
roots.other he t
are and ,, that show ,01 ofroot complex a is If b) 54325 zi
roots.other he t
are and ,, that show ,01 ofroot complex a is If b) 54325 zi
15 z 1hen solution tais If 5
roots.other he t
are and ,, that show ,01 ofroot complex a is If b) 54325 zi
15 z 1hen solution tais If 5
11555
solutionais5
roots.other he t
are and ,, that show ,01 ofroot complex a is If b) 54325 zi
15 z 1hen solution tais If 5
11555
solutionais5
2552
solutionais 2112
roots.other he t
are and ,, that show ,01 ofroot complex a is If b) 54325 zi
15 z 1hen solution tais If 5
11555
solutionais5
2552
solutionais 2112
3553
solutionais3113
roots.other he t
are and ,, that show ,01 ofroot complex a is If b) 54325 zi
15 z 1hen solution tais If 5
11555
solutionais5
2552
solutionais 2112
3553
solutionais3113
4554
solutionais4114
roots.other he t
are and ,, that show ,01 ofroot complex a is If b) 54325 zi
15 z 1hen solution tais If 5
11555
solutionais5
2552
solutionais 2112
3553
solutionais3113
4554
solutionais4114
roots also are and ,,root then a is if Thus 5432
0 1 that Prove 432 ii
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
011
01:
12 n
n
NOTE
01 432 1
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
011
01:
12 n
n
NOTE
01 432 1
OR2 3 41 GP: 1, , 5a r n 1
1
na rr
51 1
1
0 5 1 0
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
011
01:
12 n
n
NOTE
01 432 1
324 and are roots hoseequation w quadratic theFind iii
OR2 3 41 GP: 1, , 5a r n 1
1
na rr
51 1
1
0 5 1 0
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
011
01:
12 n
n
NOTE
01 432 1
324 and are roots hoseequation w quadratic theFind iii324
1 324
7643
1
243
OR2 3 41 GP: 1, , 5a r n 1
1
na rr
51 1
1
0 5 1 0
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
011
01:
12 n
n
NOTE
01 432 1
324 and are roots hoseequation w quadratic theFind iii324
1 324
7643
1
243
01 isequation 2 xx
OR2 3 41 GP: 1, , 5a r n 1
1
na rr
51 1
1
0 5 1 0
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
1
3
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
1
3
2
1 1( ) Evaluate 1 1
ii
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
1
3
2
1 1( ) Evaluate 1 1
ii
11
2
21
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
1
3
2
1 1( ) Evaluate 1 1
ii
11
2
21
1
2
2
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
1
3
2
1 1( ) Evaluate 1 1
ii
11
2
21
1
2
2
(iii) Form the cubic equation with roots 21 ,1 ,1
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
1
3
2
1 1( ) Evaluate 1 1
ii
11
2
21
1
2
2
(iii) Form the cubic equation with roots 21 ,1 ,1 0121 3222 zzz
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
1
3
2
1 1( ) Evaluate 1 1
ii
11
2
21
1
2
2
(iii) Form the cubic equation with roots 21 ,1 ,1 0121 3222 zzz
011 2 zzz
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
1
3
2
1 1( ) Evaluate 1 1
ii
11
2
21
1
2
2
(iii) Form the cubic equation with roots 21 ,1 ,1 0121 3222 zzz
011 2 zzz01223 zzzzz0122 23 zzz
5 4 3 2d) Solve 1 0z z z z z
5 4 3 2d) Solve 1 0z z z z z
111 Now 23456 zzzzzzz
5 4 3 2d) Solve 1 0z z z z z
111 Now 23456 zzzzzzz6And 1 0 has solutions;
2 0, 1, 2,36
zkz cis k
5 4 3 2d) Solve 1 0z z z z z
111 Now 23456 zzzzzzz
012345 zzzz
6And 1 0 has solutions;2 0, 1, 2,3
6
zkz cis k
5 4 3 2d) Solve 1 0z z z z z
111 Now 23456 zzzzzzz
012345 zzzz
1,016 zz
01
11 2345
z
zzzzz
6And 1 0 has solutions;2 0, 1, 2,3
6
zkz cis k
5 4 3 2d) Solve 1 0z z z z z
111 Now 23456 zzzzzzz
012345 zzzz
1,016 zz
01
11 2345
z
zzzzz
6And 1 0 has solutions;2 0, 1, 2,3
6
zkz cis k
2 2, , , ,3 3 3 3
z cis cis cis cis cis
1 3 1 3 1 3 1 3, , , , 12 2 2 2 2 2 2 2
z i i i i
9
e) 1996 HSC2 2 Let cos sin9 9
(i) Show that is a solution of 1 0, where is an integerk
i
z k
9
e) 1996 HSC2 2 Let cos sin9 9
(i) Show that is a solution of 1 0, where is an integerk
i
z k
19 z
9
e) 1996 HSC2 2 Let cos sin9 9
(i) Show that is a solution of 1 0, where is an integerk
i
z k
19 z
8,7,6,5,4,3,2,1,0 9
2
kkcisz
k
cisz
92
9
e) 1996 HSC2 2 Let cos sin9 9
(i) Show that is a solution of 1 0, where is an integerk
i
z k
19 z
8,7,6,5,4,3,2,1,0 9
2
kkcisz
k
cisz
92
kz
9
e) 1996 HSC2 2 Let cos sin9 9
(i) Show that is a solution of 1 0, where is an integerk
i
z k
19 z
8,7,6,5,4,3,2,1,0 9
2
kkcisz
k
cisz
92
kz 2 3 4 5 6 7 8(ii) Prove that 1
019 z
9
e) 1996 HSC2 2 Let cos sin9 9
(i) Show that is a solution of 1 0, where is an integerk
i
z k
19 z
8,7,6,5,4,3,2,1,0 9
2
kkcisz
k
cisz
92
kz 2 3 4 5 6 7 8(ii) Prove that 1
019 z01 8765432 rootsofsum
9
e) 1996 HSC2 2 Let cos sin9 9
(i) Show that is a solution of 1 0, where is an integerk
i
z k
19 z
8,7,6,5,4,3,2,1,0 9
2
kkcisz
k
cisz
92
kz 2 3 4 5 6 7 8(ii) Prove that 1
019 z01 8765432 rootsofsum
18765432
2 4 1(iii) Hence show that cos cos cos9 9 9 8
2 4 1(iii) Hence show that cos cos cos9 9 9 8
9
8 2 7 3 6 4 5
1
1
z
z z z z z z z z z
2 4 1(iii) Hence show that cos cos cos9 9 9 8
9
8 2 7 3 6 4 5
1
1
z
z z z z z z z z z
2 2
2 2
2 41 2cos 1 2cos 19 9
6 8 2cos 1 2cos 19 9
z z z z z
z z z z
2 2
2 2
2 41 2cos 1 2cos 19 9
8 1 2cos 19
z z z z z
z z z z
2 4 1(iii) Hence show that cos cos cos9 9 9 8
9
8 2 7 3 6 4 5
1
1
z
z z z z z z z z z
2 2
2 2
2 41 2cos 1 2cos 19 9
6 8 2cos 1 2cos 19 9
z z z z z
z z z z
2 2
2 2
2 41 2cos 1 2cos 19 9
8 1 2cos 19
z z z z z
z z z z
Let z i
9 2 4 81 1 2cos 2cos 2cos9 9 9
i i i i i i
9 2 4 81 1 2cos 2cos 2cos9 9 9
i i i i i i
9 2 4 81 1 2cos 2cos 2cos9 9 9
i i i i i i
4 2 4 81 1 2cos 2cos 2cos9 9 9
i i i
9 2 4 81 1 2cos 2cos 2cos9 9 9
i i i i i i
4 2 4 81 1 2cos 2cos 2cos9 9 9
i i i
2 4 81 8cos cos cos9 9 9
9 2 4 81 1 2cos 2cos 2cos9 9 9
i i i i i i
4 2 4 81 1 2cos 2cos 2cos9 9 9
i i i
2 4 81 8cos cos cos9 9 9
2 41 8cos cos cos9 9 9
9 2 4 81 1 2cos 2cos 2cos9 9 9
i i i i i i
4 2 4 81 1 2cos 2cos 2cos9 9 9
i i i
2 4 81 8cos cos cos9 9 9
2 4 1cos cos cos9 9 9 8
2 41 8cos cos cos9 9 9
9 2 4 81 1 2cos 2cos 2cos9 9 9
i i i i i i
4 2 4 81 1 2cos 2cos 2cos9 9 9
i i i
2 4 81 8cos cos cos9 9 9
2 4 1cos cos cos9 9 9 8
Cambridge: Exercise 7C; 1 to 4, 5ac, 6, 7, 8, 9, 11
Patel: Exercise 4I; 1, 3, 5 (need 4b), 7
Patel: Exercise 4J; 1 to 4, 7ac
2 41 8cos cos cos9 9 9