11X1 T08 07 asinx + bcosx = c (2010)

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  • Equations of the form asinx + bcosx = cMethod 1: Using the t results

  • Equations of the form asinx + bcosx = cMethod 1: Using the t results

    3600 eg (i) 3cos 4sin 2

  • Equations of the form asinx + bcosx = cMethod 1: Using the t results

    3600 eg (i) 3cos 4sin 2 let tan

    2t

  • Equations of the form asinx + bcosx = cMethod 1: Using the t results

    3600 eg (i) 3cos 4sin 2 2

    2 21 23 4 2 0 1801 1 2

    t tt t

    let tan

    2t

  • Equations of the form asinx + bcosx = cMethod 1: Using the t results

    3600 eg (i) 3cos 4sin 2 2

    2 21 23 4 2 0 1801 1 2

    t tt t

    let tan

    2t

    2 23 3 8 2 2t t t

  • Equations of the form asinx + bcosx = cMethod 1: Using the t results

    3600 eg (i) 3cos 4sin 2 2

    2 21 23 4 2 0 1801 1 2

    t tt t

    let tan

    2t

    2 23 3 8 2 2t t t 25 8 1 0t t

  • Equations of the form asinx + bcosx = cMethod 1: Using the t results

    3600 eg (i) 3cos 4sin 2 2

    2 21 23 4 2 0 1801 1 2

    t tt t

    let tan

    2t

    2 23 3 8 2 2t t t 25 8 1 0t t

    8 8410

    t

  • Equations of the form asinx + bcosx = cMethod 1: Using the t results

    3600 eg (i) 3cos 4sin 2 2

    2 21 23 4 2 0 1801 1 2

    t tt t

    let tan

    2t

    2 23 3 8 2 2t t t 25 8 1 0t t

    8 8410

    t 4 21 4 21tan or tan

    2 5 2 5

  • Equations of the form asinx + bcosx = cMethod 1: Using the t results

    3600 eg (i) 3cos 4sin 2 2

    2 21 23 4 2 0 1801 1 2

    t tt t

    let tan

    2t

    2 23 3 8 2 2t t t 25 8 1 0t t

    8 8410

    t 4 21 4 21tan or tan

    2 5 2 5

    Q2

  • Equations of the form asinx + bcosx = cMethod 1: Using the t results

    3600 eg (i) 3cos 4sin 2 2

    2 21 23 4 2 0 1801 1 2

    t tt t

    let tan

    2t

    2 23 3 8 2 2t t t 25 8 1 0t t

    8 8410

    t 4 21 4 21tan or tan

    2 5 2 5

    Q2 21 4tan5

    6 39

  • Equations of the form asinx + bcosx = cMethod 1: Using the t results

    3600 eg (i) 3cos 4sin 2 2

    2 21 23 4 2 0 1801 1 2

    t tt t

    let tan

    2t

    2 23 3 8 2 2t t t 25 8 1 0t t

    8 8410

    t 4 21 4 21tan or tan

    2 5 2 5

    Q2 21 4tan5

    6 39

    173 212

    346 42

  • Equations of the form asinx + bcosx = cMethod 1: Using the t results

    3600 eg (i) 3cos 4sin 2 2

    2 21 23 4 2 0 1801 1 2

    t tt t

    let tan

    2t

    2 23 3 8 2 2t t t 25 8 1 0t t

    8 8410

    t 4 21 4 21tan or tan

    2 5 2 5

    Q2 21 4tan5

    6 39

    173 212

    346 42

    Q1

  • Equations of the form asinx + bcosx = cMethod 1: Using the t results

    3600 eg (i) 3cos 4sin 2 2

    2 21 23 4 2 0 1801 1 2

    t tt t

    let tan

    2t

    2 23 3 8 2 2t t t 25 8 1 0t t

    8 8410

    t 4 21 4 21tan or tan

    2 5 2 5

    Q2 21 4tan5

    6 39

    173 212

    346 42

    Q1 4 21tan5

    59 47

  • Equations of the form asinx + bcosx = cMethod 1: Using the t results

    3600 eg (i) 3cos 4sin 2 2

    2 21 23 4 2 0 1801 1 2

    t tt t

    let tan

    2t

    2 23 3 8 2 2t t t 25 8 1 0t t

    8 8410

    t 4 21 4 21tan or tan

    2 5 2 5

    Q2 21 4tan5

    6 39

    173 212

    346 42

    Q1 4 21tan5

    59 47

    59 472

    119 33

  • Equations of the form asinx + bcosx = cMethod 1: Using the t results

    3600 eg (i) 3cos 4sin 2 2

    2 21 23 4 2 0 1801 1 2

    t tt t

    let tan

    2t

    2 23 3 8 2 2t t t 25 8 1 0t t

    8 8410

    t 4 21 4 21tan or tan

    2 5 2 5

    Q2 21 4tan5

    6 39

    173 212

    346 42

    Q1 4 21tan5

    59 47

    59 472

    119 33

    180: Test

  • Equations of the form asinx + bcosx = cMethod 1: Using the t results

    3600 eg (i) 3cos 4sin 2 2

    2 21 23 4 2 0 1801 1 2

    t tt t

    let tan

    2t

    2 23 3 8 2 2t t t 25 8 1 0t t

    8 8410

    t 4 21 4 21tan or tan

    2 5 2 5

    Q2 21 4tan5

    6 39

    173 212

    346 42

    Q1 4 21tan5

    59 47

    59 472

    119 33

    180: Test3cos180 4sin180

    4 2

  • Equations of the form asinx + bcosx = cMethod 1: Using the t results

    3600 eg (i) 3cos 4sin 2 2

    2 21 23 4 2 0 1801 1 2

    t tt t

    let tan

    2t

    2 23 3 8 2 2t t t 25 8 1 0t t

    8 8410

    t 4 21 4 21tan or tan

    2 5 2 5

    Q2 21 4tan5

    6 39

    173 212

    346 42

    Q1 4 21tan5

    59 47

    59 472

    119 33

    180: Test3cos180 4sin180

    4 2

    119 33 ,346 42

  • Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600

  • Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2

  • Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 sincoscossin

  • Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 sincoscossin

  • Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 sincoscossin

    sin corresponds to 3, so 3 goes on the opposite

    side

    3

  • Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 sincoscossin

    3

  • Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 sincoscossin

    3

    cos corresponds to 4, so 4 goes on the adjacent

    side

    4

  • Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 sincoscossin

    3

    4

    by Pythagoras the hypotenuse is 5

    5

  • Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 sincoscossin

    3

    4

    by Pythagoras the hypotenuse is 5

    5

    3 45 cos sin 25 5

  • Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 sincoscossin

    3

    4

    by Pythagoras the hypotenuse is 5

    5

    5sin 2 3 45 cos sin 25 5

  • Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 sincoscossin

    3

    4

    by Pythagoras the hypotenuse is 5

    5

    5sin 2

    The hypotenuse becomes the coefficient of the trig function

    3 45 cos sin 25 5

  • Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 3

    4

    5

    5sin 2 3 45 cos sin 25 5

    3tan4

    2sin

    5

  • Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 3

    4

    5

    5sin 2 3 45 cos sin 25 5

    3tan4

    36 52 2sin

    5

  • Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 3

    4

    5

    5sin 2 3 45 cos sin 25 5

    3tan4

    36 52 2sin

    5 Q1, Q2

  • Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 3

    4

    5

    5sin 2 3 45 cos sin 25 5

    3tan4

    36 52 2sin

    5 Q1, Q2

    2sin5

  • Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 3

    4

    5

    5sin 2 3 45 cos sin 25 5

    3tan4

    36 52 2sin

    5 Q1, Q2

    2sin5

    23 35

  • Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 3

    4

    5

    5sin 2 3 45 cos sin 25 5

    3tan4

    36 52 2sin

    5 Q1, Q2

    2sin5

    23 35

    36 52 23 35 ,156 25

  • Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 3

    4

    5

    5sin 2 3 45 cos sin 25 5

    3tan4

    36 52 2sin

    5 Q1, Q2

    2sin5

    23 35

    36 52 23 35 ,156 25 13 17 ,119 33

  • Method 2: Auxiliary Angle Method(i) Change into a sine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 3

    4

    5

    5sin 2 3 45 cos sin 25 5

    3tan4

    36 52 2sin

    5 Q1, Q2

    2sin5

    23 35

    36 52 23 35 ,156 25 119 33 ,346 43 13 17 ,119 33

  • Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600

  • Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2

  • Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 cos cos sin sin

  • Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 cos cos sin sin

  • Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 cos cos sin sin

    cos corresponds to 3, so 3 goes on the adjacent

    side

    3

  • Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 cos cos sin sin

    3

  • Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 cos cos sin sin

    3

    cos corresponds to 4, so 4 goes on the adjacent

    side

    4

  • Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 cos cos sin sin

    3

    4

    by Pythagoras the hypotenuse is 5

    5

  • Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 cos cos sin sin

    3

    4

    by Pythagoras the hypotenuse is 5

    5

    3 45 cos sin 25 5

  • Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 cos cos sin sin

    3

    4

    by Pythagoras the hypotenuse is 5

    5

    5cos 2 3 45 cos sin 25 5

  • Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 cos cos sin sin

    3

    4

    by Pythagoras the hypotenuse is 5

    5

    5cos 2

    The hypotenuse becomes the coefficient of the trig function

    3 45 cos sin 25 5

  • Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 cos cos sin sin

    3

    45

    5cos 2 3 45 cos sin 25 5

    4tan3

    2cos

    5

  • Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 cos cos sin sin

    3

    45

    5cos 2 3 45 cos sin 25 5

    4tan3

    53 8 2cos

    5

  • Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 cos cos sin sin

    3

    45

    5cos 2 3 45 cos sin 25 5

    4tan3

    53 8 2cos

    5 Q1, Q4

  • Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 cos cos sin sin

    3

    45

    5cos 2 3 45 cos sin 25 5

    4tan3

    53 8 2cos

    5 Q1, Q4

    2cos5

  • Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 cos cos sin sin

    3

    45

    5cos 2 3 45 cos sin 25 5

    4tan3

    53 8 2cos

    5 Q1, Q4

    2cos5

    66 25

  • Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 cos cos sin sin

    3

    45

    5cos 2 3 45 cos sin 25 5

    4tan3

    53 8 2cos

    5 Q1, Q4

    2cos5

    66 25

    53 8 66 25 ,293 35

  • Method 2: Auxiliary Angle Method(ii) Change into a cosine functioneg (i) 3cos 4sin 2 3600

    3cos 4sin 2 cos cos sin sin

    3

    45

    5cos 2 3 45 cos sin 25 5

    4tan3

    53 8 2cos

    5 Q1, Q4

    2cos5

    66 25

    53 8 66 25 ,293 35 119 33 ,346 43

  • (ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)

  • (ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)

    3sin3 cos3t t

  • (ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)

    3sin3 cos3t t sin 3 cos cos3 sint t

  • (ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)

    3sin3 cos3t t sin 3 cos cos3 sint t

    3

    12

  • (ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)

    3sin3 cos3t t sin 3 cos cos3 sint t

    3

    12 2sin 3t

  • (ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)

    3sin3 cos3t t sin 3 cos cos3 sint t

    3

    12 2sin 3t 1tan3

    30

  • (ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)

    3sin3 cos3t t sin 3 cos cos3 sint t

    3

    12 2sin 3t 1tan3

    30

    2sin 3 30t

  • (ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)

    3sin3 cos3t t sin 3 cos cos3 sint t

    3

    12 2sin 3t 1tan3

    30

    2sin 3 30t

    (iii) Express sin cos in the form cosx x R x 2003 Extension 1 HSC Q2e) (i)

  • (ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)

    3sin3 cos3t t sin 3 cos cos3 sint t

    3

    12 2sin 3t 1tan3

    30

    2sin 3 30t

    (iii) Express sin cos in the form cosx x R x 2003 Extension 1 HSC Q2e) (i)

    sin cosx x

  • (ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)

    3sin3 cos3t t sin 3 cos cos3 sint t

    3

    12 2sin 3t 1tan3

    30

    2sin 3 30t

    (iii) Express sin cos in the form cosx x R x 2003 Extension 1 HSC Q2e) (i)

    sin cosx x cos cos sin sinx x

  • (ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)

    3sin3 cos3t t sin 3 cos cos3 sint t

    3

    12 2sin 3t 1tan3

    30

    2sin 3 30t

    (iii) Express sin cos in the form cosx x R x 2003 Extension 1 HSC Q2e) (i)

    sin cosx x cos cos sin sinx x

    cos sinx x

  • (ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)

    3sin3 cos3t t sin 3 cos cos3 sint t

    3

    12 2sin 3t 1tan3

    30

    2sin 3 30t

    (iii) Express sin cos in the form cosx x R x 2003 Extension 1 HSC Q2e) (i)

    sin cosx x cos cos sin sinx x

    1

    12 cos sinx x

  • (ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)

    3sin3 cos3t t sin 3 cos cos3 sint t

    3

    12 2sin 3t 1tan3

    30

    2sin 3 30t

    (iii) Express sin cos in the form cosx x R x 2003 Extension 1 HSC Q2e) (i)

    sin cosx x cos cos sin sinx x

    1

    12

    2 cos x cos sinx x

  • (ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)

    3sin3 cos3t t sin 3 cos cos3 sint t

    3

    12 2sin 3t 1tan3

    30

    2sin 3 30t

    (iii) Express sin cos in the form cosx x R x 2003 Extension 1 HSC Q2e) (i)

    sin cosx x cos cos sin sinx x

    1

    12

    2 cos x tan 1

    45

    cos sinx x

  • (ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)

    3sin3 cos3t t sin 3 cos cos3 sint t

    3

    12 2sin 3t 1tan3

    30

    2sin 3 30t

    (iii) Express sin cos in the form cosx x R x 2003 Extension 1 HSC Q2e) (i)

    sin cosx x cos cos sin sinx x

    1

    12

    2 cos x tan 1

    45

    2 cos 45x cos sinx x

  • (ii) Express 3sin 3 cos3 in the form sin 3t t R t 2005 Extension 1 HSC Q5c) (i)

    3sin3 cos3t t sin 3 cos cos3 sint t

    3

    12 2sin 3t 1tan3

    30

    2sin 3 30t

    (iii) Express sin cos in the form cosx x R x 2003 Extension 1 HSC Q2e) (i)

    sin cosx x cos cos sin sinx x

    1

    12

    2 cos x tan 1

    45

    2 cos 45x cos sinx x

    Exercise 2E;6, 7, 10bd, 11, 13, 14, 16ac, 20a, 23

    Slide Number 1Slide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34Slide Number 35Slide Number 36Slide Number 37Slide Number 38Slide Number 39Slide Number 40Slide Number 41Slide Number 42Slide Number 43Slide Number 44Slide Number 45Slide Number 46Slide Number 47Slide Number 48Slide Number 49Slide Number 50Slide Number 51Slide Number 52Slide Number 53Slide Number 54Slide Number 55Slide Number 56Slide Number 57Slide Number 58Slide Number 59Slide Number 60Slide Number 61Slide Number 62Slide Number 63Slide Number 64Slide Number 65Slide Number 66Slide Number 67Slide Number 68Slide Number 69Slide Number 70