x2 t01 10 complex & trig (2013)
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Complex Numbers & Trig Identities
(i) Express cos 2 and sin 2 in terms of cos and sin
Complex Numbers & Trig Identities
(i) Express cos 2 and sin 2 in terms of cos and sin
2sincos2sin2cos ii
Complex Numbers & Trig Identities
(i) Express cos 2 and sin 2 in terms of cos and sin
2sincos2sin2cos ii 22 sincossin2cos i
Complex Numbers & Trig Identities
(i) Express cos 2 and sin 2 in terms of cos and sin
2sincos2sin2cos ii 22 sincossin2cos i
By equating real and imaginary parts
Complex Numbers & Trig Identities
22 sincos2cos
(i) Express cos 2 and sin 2 in terms of cos and sin
2sincos2sin2cos ii 22 sincossin2cos i
cossin22sin
By equating real and imaginary parts
(ii) Express sin3 ,cos3 and tan3 in terms of sin ,cos and tan
(ii) Express sin3 ,cos3 and tan3 in terms of sin ,cos and tan
3cos3 sin3 cos sini i
(ii) Express sin3 ,cos3 and tan3 in terms of sin ,cos and tan
3cos3 sin3 cos sini i 3 2 2 3cos 3 sin cos 3sin cos sini i
(ii) Express sin3 ,cos3 and tan3 in terms of sin ,cos and tan
3cos3 sin3 cos sini i
(ii) Express sin3 ,cos3 and tan3 in terms of sin ,cos and tan
3cos3 sin3 cos sini i 3 2 2 3cos 3 sin cos 3sin cos sini i
2 3sin3 3sin cos sin
By equating real and imaginary parts
(ii) Express sin3 ,cos3 and tan3 in terms of sin ,cos and tan
3cos3 sin3 cos sini i 3 2 2 3cos 3 sin cos 3sin cos sini i
2 3sin3 3sin cos sin
By equating real and imaginary parts
2 3
3 3
3
3sin 1 sin sin
3sin 3sin sin3sin 4sin
3 2cos3 cos 3sin cos
(ii) Express sin3 ,cos3 and tan3 in terms of sin ,cos and tan
3cos3 sin3 cos sini i 3 2 2 3cos 3 sin cos 3sin cos sini i
2 3sin3 3sin cos sin
By equating real and imaginary parts
2 3
3 3
3
3sin 1 sin sin
3sin 3sin sin3sin 4sin
3 2cos3 cos 3sin cos
(ii) Express sin3 ,cos3 and tan3 in terms of sin ,cos and tan
3cos3 sin3 cos sini i 3 2 2 3cos 3 sin cos 3sin cos sini i
2 3sin3 3sin cos sin
By equating real and imaginary parts
2 3
3 3
3
3sin 1 sin sin
3sin 3sin sin3sin 4sin
3 2
3 3
3
cos 3 1 cos cos
cos 3cos 3cos4cos 3cos
3 2cos3 cos 3sin cos
(ii) Express sin3 ,cos3 and tan3 in terms of sin ,cos and tan
3cos3 sin3 cos sini i 3 2 2 3cos 3 sin cos 3sin cos sini i
2 3sin3 3sin cos sin
By equating real and imaginary parts
2 3
3 3
3
3sin 1 sin sin
3sin 3sin sin3sin 4sin
3 2
3 3
3
cos 3 1 cos cos
cos 3cos 3cos4cos 3cos
sin3tan3cos3
2 3
3 2
3sin cos sincos 3sin cos
3 2cos3 cos 3sin cos
(ii) Express sin3 ,cos3 and tan3 in terms of sin ,cos and tan
3cos3 sin3 cos sini i 3 2 2 3cos 3 sin cos 3sin cos sini i
2 3sin3 3sin cos sin
By equating real and imaginary parts
2 3
3 3
3
3sin 1 sin sin
3sin 3sin sin3sin 4sin
3 2
3 3
3
cos 3 1 cos cos
cos 3cos 3cos4cos 3cos
sin3tan3cos3
2 3
3 2
3sin cos sincos 3sin cos
2 3
3
3 2
3
3sin cos sincos
cos 3sin coscos
3
3
3 2
3 2
3sin sincos cos
cos 3sincos cos
3 2cos3 cos 3sin cos
(ii) Express sin3 ,cos3 and tan3 in terms of sin ,cos and tan
3cos3 sin3 cos sini i 3 2 2 3cos 3 sin cos 3sin cos sini i
2 3sin3 3sin cos sin
By equating real and imaginary parts
2 3
3 3
3
3sin 1 sin sin
3sin 3sin sin3sin 4sin
3 2
3 3
3
cos 3 1 cos cos
cos 3cos 3cos4cos 3cos
sin3tan3cos3
2 3
3 2
3sin cos sincos 3sin cos
2 3
3
3 2
3
3sin cos sincos
cos 3sin coscos
3
3
3 2
3 2
3sin sincos cos
cos 3sincos cos
3
2
3tan tan1 3tan
3(iii) Show that cos is a solution to 8 6 1 09
x x x
3(iii) Show that cos is a solution to 8 6 1 09
x x x
3
3
8 6 1 02(4 3 ) 1 0x x
x x
3(iii) Show that cos is a solution to 8 6 1 09
x x x
3
3
8 6 1 02(4 3 ) 1 0x x
x x
let cosx
3(iii) Show that cos is a solution to 8 6 1 09
x x x
3
3
8 6 1 02(4 3 ) 1 0x x
x x
let cosx 32(4cos 3cos ) 1 0
2cos3 1 01cos32
3(iii) Show that cos is a solution to 8 6 1 09
x x x
3
3
8 6 1 02(4 3 ) 1 0x x
x x
let cosx 32(4cos 3cos ) 1 0
2cos3 1 01cos32
5 73 , ,3 3 3
5 7, ,9 9 9
3(iii) Show that cos is a solution to 8 6 1 09
x x x
3
3
8 6 1 02(4 3 ) 1 0x x
x x
let cosx 32(4cos 3cos ) 1 0
2cos3 1 01cos32
5 73 , ,3 3 3
5 7, ,9 9 9
5 7cos ,cos ,sin9 9 9
x
3(iii) Show that cos is a solution to 8 6 1 09
x x x
3
3
8 6 1 02(4 3 ) 1 0x x
x x
let cosx 32(4cos 3cos ) 1 0
2cos3 1 01cos32
5 73 , ,3 3 3
5 7, ,9 9 9
5 7cos ,cos ,sin9 9 9
x cos is a solution
9x
2 5(iii) Evaluate cos cos cos9 9 9
2 5(iii) Evaluate cos cos cos9 9 9
35 7cos ,cos ,sin are the roots of 8 6 1 09 9 9
x x x
2 5(iii) Evaluate cos cos cos9 9 9
35 7cos ,cos ,sin are the roots of 8 6 1 09 9 9
x x x
18
product of roots
5 7 1cos cos cos9 9 9 8
2 5(iii) Evaluate cos cos cos9 9 9
35 7cos ,cos ,sin are the roots of 8 6 1 09 9 9
x x x
18
product of roots
5 7 1cos cos cos9 9 9 8
7 2but cos cos9 9
2 5(iii) Evaluate cos cos cos9 9 9
35 7cos ,cos ,sin are the roots of 8 6 1 09 9 9
x x x
18
product of roots
5 7 1cos cos cos9 9 9 8
7 2but cos cos9 9
2 5 1cos cos cos9 9 9 8
1 1 If cos sin , find and n nn niv z i z z
z z
1 1 If cos sin , find and n nn niv z i z z
z z
ninzn sincos
1 1 If cos sin , find and n nn niv z i z z
z z
ninzn sincos
ninzz
nn sincos1
1 1 If cos sin , find and n nn niv z i z z
z z
ninzn sincos
ninzz
nn sincos1
nin sincos
xxxx
sinsinfunction odd is sincoscosfunctioneven is cos
1 1 If cos sin , find and n nn niv z i z z
z z
ninzn sincos
ninzz
nn sincos1
nin sincos
xxxx
sinsinfunction odd is sincoscosfunctioneven is cos
nz
z nn cos21
niz
z nn sin21
1 1 If cos sin , find and n nn niv z i z z
z z
ninzn sincos
ninzz
nn sincos1
nin sincos
xxxx
sinsinfunction odd is sincoscosfunctioneven is cos
nz
z nn cos21
niz
z nn sin21
3 Express cos in terms of cosv n
1 1 If cos sin , find and n nn niv z i z z
z z
ninzn sincos
ninzz
nn sincos1
nin sincos
xxxx
sinsinfunction odd is sincoscosfunctioneven is cos
nz
z nn cos21
niz
z nn sin21
3 Express cos in terms of cosv n
3
3 1cos2
zz
333 133cos8
zzzz
1 1 If cos sin , find and n nn niv z i z z
z z
ninzn sincos
ninzz
nn sincos1
nin sincos
xxxx
sinsinfunction odd is sincoscosfunctioneven is cos
nz
z nn cos21
niz
z nn sin21
3 Express cos in terms of cosv n
3
3 1cos2
zz
333 133cos8
zzzz
zz
zz 131
33
1 1 If cos sin , find and n nn niv z i z z
z z
ninzn sincos
ninzz
nn sincos1
nin sincos
xxxx
sinsinfunction odd is sincoscosfunctioneven is cos
nz
z nn cos21
niz
z nn sin21
3 Express cos in terms of cosv n
3
3 1cos2
zz
333 133cos8
zzzz
zz
zz 131
33
38cos 2cos3 6cos
cos433cos
41cos3
Cambridge: Exercise 7B; 2 to 5, 7 to 12
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