11 x1 t08 04 double angles (2012)

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  • Double Angles

  • Double Angles sin 2 sin

  • Double Angles sin 2 sin

    sin cos cos sin

  • Double Angles sin 2 sin

    sin cos cos sin sin 2 2sin cos

  • Double Angles sin 2 sin

    sin cos cos sin sin 2 2sin cos

    cos 2 cos

  • Double Angles sin 2 sin

    sin cos cos sin sin 2 2sin cos

    cos 2 cos cos cos sin sin

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • Double Angles cossin22sin

  • Double Angles cossin22sin

    22 sincos2cos

  • Double Angles cossin22sin

    22 sincos2cos

    1cos2 2

  • Double Angles cossin22sin

    22 sincos2cos

    1cos2 2 2cos121cos2

  • Double Angles cossin22sin

    22 sincos2cos

    1cos2 2 2cos121cos2

    2sin21

  • Double Angles cossin22sin

    22 sincos2cos

    1cos2 2 2cos121cos2

    2sin21 2cos121sin2

  • Double Angles cossin22sin

    22 sincos2cos

    1cos2 2 2cos121cos2

    2sin21 2cos121sin2

    2tan1

    tan22tan

  • Double Angles cossin22sin

    22 sincos2cos

    1cos2 2 2cos121cos2

    2sin21 2cos121sin2

    2tan1

    tan22tan

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

  • Double Angles cossin22sin

    22 sincos2cos

    1cos2 2 2cos121cos2

    2sin21 2cos121sin2

    2tan1

    tan22tan

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

    2

    3 5

  • 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

  • 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

  • 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

  • 5 5ii Find the exact value of sin cos12 12

  • 5 5ii Find the exact value of sin cos12 12

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

    2 12 12

  • 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

  • 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

  • 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

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

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

    2 1sin 1 cos 22

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

    2 1sin 1 cos 22

    2 1sin 1 cos2 2

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

    2 1sin 1 cos 22

    2 1sin 1 cos2 2

    1 212 3

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

    2 1sin 1 cos 22

    2 1sin 1 cos2 2

    1 212 3

    16

  • 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

  • 1 cos 2iv Prove tan1 cos 2

    x xx

  • 1 cos 2iv Prove tan1 cos 2

    x xx

    1 cos 21 cos 2

    xx

    2

    2

    1 1 2sin1 2cos 1

    xx

  • 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

  • 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

  • 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

  • 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

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

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

    sin3 cos3sin cos

    sin3 cos cos3 sin

    sin cos

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

    sin3 cos3sin cos

    cossin23sin2

    sin3 cos cos3 sinsin cos

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

    sin3 cos3sin cos

    cossin23sin2

    2sin2sin2

    sin3 cos cos3 sinsin cos

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

    sin3 cos3sin cos

    cossin23sin2

    2sin2sin2

    2

    sin3 cos cos3 sinsin cos

  • 1994 Extension 1 HSC Q2a)

    2

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

    1 tanA AA

  • 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

  • 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

  • 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

  • 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

  • 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*

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