11 x1 t08 04 double angles (2012)

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Double Angles

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Page 1: 11 x1 t08 04 double angles (2012)

Double Angles

Page 2: 11 x1 t08 04 double angles (2012)

Double Angles sin 2 sin

Page 3: 11 x1 t08 04 double angles (2012)

Double Angles sin 2 sin

sin cos cos sin

Page 4: 11 x1 t08 04 double angles (2012)

Double Angles sin 2 sin

sin cos cos sin

sin 2 2sin cos

Page 5: 11 x1 t08 04 double angles (2012)

Double Angles sin 2 sin

sin cos cos sin

sin 2 2sin cos

cos 2 cos

Page 6: 11 x1 t08 04 double angles (2012)

Double Angles sin 2 sin

sin cos cos sin

sin 2 2sin cos

cos 2 cos cos cos sin sin

Page 7: 11 x1 t08 04 double angles (2012)

Double Angles sin 2 sin

sin cos cos sin

sin 2 2sin cos

cos 2 cos cos cos sin sin

2 2cos 2 cos sin

Page 8: 11 x1 t08 04 double angles (2012)

Double Angles sin 2 sin

sin cos cos sin

sin 2 2sin cos

cos 2 cos cos cos sin sin

2 2cos 2 cos sin

2 2cos 1 cos

Page 9: 11 x1 t08 04 double angles (2012)

Double Angles sin 2 sin

sin cos cos sin

sin 2 2sin cos

cos 2 cos cos cos sin sin

2 2cos 2 cos sin

2 2cos 1 cos 2cos 2 2cos 1

Page 10: 11 x1 t08 04 double angles (2012)

Double Angles sin 2 sin

sin cos cos sin

sin 2 2sin cos

cos 2 cos cos cos sin sin

2 2cos 2 cos sin

2 2cos 1 cos 2cos 2 2cos 1

22 1 sin 1

Page 11: 11 x1 t08 04 double angles (2012)

Double Angles sin 2 sin

sin cos cos sin

sin 2 2sin cos

cos 2 cos cos cos sin sin

2 2cos 2 cos sin

2 2cos 1 cos 2cos 2 2cos 1

22 1 sin 1 2cos 2 1 2sin

Page 12: 11 x1 t08 04 double angles (2012)

Double Angles sin 2 sin

sin cos cos sin

sin 2 2sin cos

cos 2 cos cos cos sin sin

2 2cos 2 cos sin

2 2cos 1 cos 2cos 2 2cos 1

22 1 sin 1 2cos 2 1 2sin

tan 2 tan

Page 13: 11 x1 t08 04 double angles (2012)

Double Angles sin 2 sin

sin cos cos sin

sin 2 2sin cos

cos 2 cos cos cos sin sin

2 2cos 2 cos sin

2 2cos 1 cos 2cos 2 2cos 1

22 1 sin 1 2cos 2 1 2sin

tan 2 tan tan tan

1 tan tan

Page 14: 11 x1 t08 04 double angles (2012)

Double Angles sin 2 sin

sin cos cos sin

sin 2 2sin cos

cos 2 cos cos cos sin sin

2 2cos 2 cos sin

2 2cos 1 cos 2cos 2 2cos 1

22 1 sin 1 2cos 2 1 2sin

tan 2 tan tan tan

1 tan tan

22 tantan 2

1 tan

Page 15: 11 x1 t08 04 double angles (2012)

Double Angles cossin22sin

Page 16: 11 x1 t08 04 double angles (2012)

Double Angles cossin22sin

22 sincos2cos

Page 17: 11 x1 t08 04 double angles (2012)

Double Angles cossin22sin

22 sincos2cos

1cos2 2

Page 18: 11 x1 t08 04 double angles (2012)

Double Angles cossin22sin

22 sincos2cos

1cos2 2 2cos121cos2

Page 19: 11 x1 t08 04 double angles (2012)

Double Angles cossin22sin

22 sincos2cos

1cos2 2 2cos121cos2

2sin21

Page 20: 11 x1 t08 04 double angles (2012)

Double Angles cossin22sin

22 sincos2cos

1cos2 2 2cos121cos2

2sin21 2cos121sin2

Page 21: 11 x1 t08 04 double angles (2012)

Double Angles cossin22sin

22 sincos2cos

1cos2 2 2cos121cos2

2sin21 2cos121sin2

2tan1

tan22tan

Page 22: 11 x1 t08 04 double angles (2012)

Double Angles cossin22sin

22 sincos2cos

1cos2 2 2cos121cos2

2sin21 2cos121sin2

2tan1

tan22tan

2e.g. i If cos , find tan 23

Page 23: 11 x1 t08 04 double angles (2012)

Double Angles cossin22sin

22 sincos2cos

1cos2 2 2cos121cos2

2sin21 2cos121sin2

2tan1

tan22tan

2e.g. i If cos , find tan 23

2

3 5

Page 24: 11 x1 t08 04 double angles (2012)

Double Angles cossin22sin

22 sincos2cos

1cos2 2 2cos121cos2

2sin21 2cos121sin2

2tan1

tan22tan

2e.g. i If cos , find tan 23

2

3 5

2tan1

tan22tan

Page 25: 11 x1 t08 04 double angles (2012)

Double Angles cossin22sin

22 sincos2cos

1cos2 2 2cos121cos2

2sin21 2cos121sin2

2tan1

tan22tan

2e.g. i If cos , find tan 23

2

3 5

2tan1

tan22tan

2

522

tan 251

2

Page 26: 11 x1 t08 04 double angles (2012)

Double Angles cossin22sin

22 sincos2cos

1cos2 2 2cos121cos2

2sin21 2cos121sin2

2tan1

tan22tan

2e.g. i If cos , find tan 23

2

3 5

2tan1

tan22tan

2

522

tan 251

2

514

4 5

Page 27: 11 x1 t08 04 double angles (2012)

5 5ii Find the exact value of sin cos12 12

Page 28: 11 x1 t08 04 double angles (2012)

5 5ii Find the exact value of sin cos12 12

5 5sin cos12 12 1 5 5= 2sin cos

2 12 12

Page 29: 11 x1 t08 04 double angles (2012)

5 5ii Find the exact value of sin cos12 12

5 5sin cos12 12 1 5 5= 2sin cos

2 12 12

1 5= sin 22 12

Page 30: 11 x1 t08 04 double angles (2012)

5 5ii Find the exact value of sin cos12 12

5 5sin cos12 12 1 5 5= 2sin cos

2 12 12

1 5= sin 22 12

1 5= sin2 6

Page 31: 11 x1 t08 04 double angles (2012)

5 5ii Find the exact value of sin cos12 12

5 5sin cos12 12 1 5 5= 2sin cos

2 12 12

1 5= sin 22 12

1 5= sin2 6

1 1=2 2

1=4

Page 32: 11 x1 t08 04 double angles (2012)

2iii If cos , find the exact value of sin3 2

Page 33: 11 x1 t08 04 double angles (2012)

2iii If cos , find the exact value of sin3 2

2 1sin 1 cos 22

Page 34: 11 x1 t08 04 double angles (2012)

2iii If cos , find the exact value of sin3 2

2 1sin 1 cos 22

2 1sin 1 cos2 2

Page 35: 11 x1 t08 04 double angles (2012)

2iii If cos , find the exact value of sin3 2

2 1sin 1 cos 22

2 1sin 1 cos2 2

1 212 3

Page 36: 11 x1 t08 04 double angles (2012)

2iii If cos , find the exact value of sin3 2

2 1sin 1 cos 22

2 1sin 1 cos2 2

1 212 3

16

Page 37: 11 x1 t08 04 double angles (2012)

2iii If cos , find the exact value of sin3 2

2 1sin 1 cos 22

2 1sin 1 cos2 2

1 212 3

16

1sin2 6

Page 38: 11 x1 t08 04 double angles (2012)

1 cos 2iv Prove tan1 cos 2

x xx

Page 39: 11 x1 t08 04 double angles (2012)

1 cos 2iv Prove tan1 cos 2

x xx

1 cos 21 cos 2

xx

2

2

1 1 2sin1 2cos 1

xx

Page 40: 11 x1 t08 04 double angles (2012)

1 cos 2iv Prove tan1 cos 2

x xx

1 cos 21 cos 2

xx

2

2

1 1 2sin1 2cos 1

xx

2

22sin2cos

xx

Page 41: 11 x1 t08 04 double angles (2012)

1 cos 2iv Prove tan1 cos 2

x xx

1 cos 21 cos 2

xx

2

2

1 1 2sin1 2cos 1

xx

2

22sin2cos

xx

2

2sincos

xx

Page 42: 11 x1 t08 04 double angles (2012)

1 cos 2iv Prove tan1 cos 2

x xx

1 cos 21 cos 2

xx

2

2

1 1 2sin1 2cos 1

xx

2

22sin2cos

xx

2

2sincos

xx

2tan x

Page 43: 11 x1 t08 04 double angles (2012)

1 cos 2iv Prove tan1 cos 2

x xx

1 cos 21 cos 2

xx

2

2

1 1 2sin1 2cos 1

xx

2

22sin2cos

xx

2

2sincos

xx

2tan x

tan x

Page 44: 11 x1 t08 04 double angles (2012)

1996 Extension 1 HSC Q4a)sin3 cos3(v) Prove that 2sin cos

Page 45: 11 x1 t08 04 double angles (2012)

1996 Extension 1 HSC Q4a)sin3 cos3(v) Prove that 2sin cos

sin3 cos3sin cos

sin3 cos cos3 sinsin cos

Page 46: 11 x1 t08 04 double angles (2012)

1996 Extension 1 HSC Q4a)sin3 cos3(v) Prove that 2sin cos

sin3 cos3sin cos

cossin23sin2

sin3 cos cos3 sinsin cos

Page 47: 11 x1 t08 04 double angles (2012)

1996 Extension 1 HSC Q4a)sin3 cos3(v) Prove that 2sin cos

sin3 cos3sin cos

cossin23sin2

2sin2sin2

sin3 cos cos3 sinsin cos

Page 48: 11 x1 t08 04 double angles (2012)

1996 Extension 1 HSC Q4a)sin3 cos3(v) Prove that 2sin cos

sin3 cos3sin cos

cossin23sin2

2sin2sin2

2

sin3 cos cos3 sinsin cos

Page 49: 11 x1 t08 04 double angles (2012)

1994 Extension 1 HSC Q2a)

2

(vi) Prove the following identity;2 tan sin 2

1 tanA AA

Page 50: 11 x1 t08 04 double angles (2012)

1994 Extension 1 HSC Q2a)

2

(vi) Prove the following identity;2 tan sin 2

1 tanA AA

22 tan

1 tanAA 2

2

2sincossin1cos

AA

AA

Page 51: 11 x1 t08 04 double angles (2012)

1994 Extension 1 HSC Q2a)

2

(vi) Prove the following identity;2 tan sin 2

1 tanA AA

22 tan

1 tanAA

AAAA22 sincos

cossin2

2

2

2sincossin1cos

AA

AA

Page 52: 11 x1 t08 04 double angles (2012)

1994 Extension 1 HSC Q2a)

2

(vi) Prove the following identity;2 tan sin 2

1 tanA AA

22 tan

1 tanAA

AAAA22 sincos

cossin2

12sin A

2

2

2sincossin1cos

AA

AA

Page 53: 11 x1 t08 04 double angles (2012)

1994 Extension 1 HSC Q2a)

2

(vi) Prove the following identity;2 tan sin 2

1 tanA AA

22 tan

1 tanAA

AAAA22 sincos

cossin2

12sin A

A2sin

2

2

2sincossin1cos

AA

AA

Page 54: 11 x1 t08 04 double angles (2012)

1994 Extension 1 HSC Q2a)

2

(vi) Prove the following identity;2 tan sin 2

1 tanA AA

22 tan

1 tanAA

AAAA22 sincos

cossin2

12sin A

A2sin

2

2

2sincossin1cos

AA

AA

Book2Exercise 2A; 2ade, 3bde, 5adej, 7, 8adg, 10ab, 11, 13ck, 16, 19*