x2 t01 10 complex & trig (2013)
TRANSCRIPT
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Complex Numbers & Trig Identities
(i) Express cos 2 and sin 2 in terms of cos and sin
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Complex Numbers & Trig Identities
(i) Express cos 2 and sin 2 in terms of cos and sin
2sincos2sin2cos ii
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Complex Numbers & Trig Identities
(i) Express cos 2 and sin 2 in terms of cos and sin
2sincos2sin2cos ii 22 sincossin2cos i
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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
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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
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(ii) Express sin3 ,cos3 and tan3 in terms of sin ,cos and tan
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(ii) Express sin3 ,cos3 and tan3 in terms of sin ,cos and tan
3cos3 sin3 cos sini i
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(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
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(ii) Express sin3 ,cos3 and tan3 in terms of sin ,cos and tan
3cos3 sin3 cos sini i
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(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
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(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
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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
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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
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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
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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
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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
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3(iii) Show that cos is a solution to 8 6 1 09
x x x
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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
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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
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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
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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
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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
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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
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2 5(iii) Evaluate cos cos cos9 9 9
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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
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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
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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
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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
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1 1 If cos sin , find and n nn niv z i z z
z z
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1 1 If cos sin , find and n nn niv z i z z
z z
ninzn sincos
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1 1 If cos sin , find and n nn niv z i z z
z z
ninzn sincos
ninzz
nn sincos1
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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
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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
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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
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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
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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
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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
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Cambridge: Exercise 7B; 2 to 5, 7 to 12