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