11 x1 t08 02 sum & difference of angles (2012)

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  • Sum & Difference of Angles

  • Sum & Difference of Anglesy

    x

    1

  • Sum & Difference of Anglesy

    x

    1

    P

  • Sum & Difference of Anglesy

    x

    1

    P

  • Sum & Difference of Anglesy

    x

    1

    P

    1

    x

    y

  • Sum & Difference of Anglesy

    x

    1

    P

    1

    x

    y

    cos1x

    cosx

    cos ,sin

  • Sum & Difference of Anglesy

    x

    1

    P

    1

    x

    y

    cos1x

    cosx sin

    1sin

    y

    y

    cos ,sin

  • Sum & Difference of Anglesy

    x

    1

    P

    1

    x

    y

    cos1x

    cosx sin

    1sin

    y

    y

    cos ,sin

    Q cos ,sin

  • Sum & Difference of Anglesy

    x

    1

    P

    1

    x

    y

    cos1x

    cosx sin

    1sin

    y

    y

    cos ,sin

    Q cos ,sin

    By trigonometry

  • Sum & Difference of Anglesy

    x

    1

    P

    1

    x

    y

    cos1x

    cosx sin

    1sin

    y

    y

    cos ,sin

    Q cos ,sin

    By trigonometry

    2 2 21 1 2 1 1 cosPQ

  • Sum & Difference of Anglesy

    x

    1

    P

    1

    x

    y

    cos1x

    cosx sin

    1sin

    y

    y

    cos ,sin

    Q cos ,sin

    By trigonometry

    2 2 21 1 2 1 1 cosPQ 2 2 2cosPQ

  • Sum & Difference of Anglesy

    x

    1

    P

    1

    x

    y

    cos1x

    cosx sin

    1sin

    y

    y

    cos ,sin

    Q cos ,sin

    By trigonometry

    2 2 21 1 2 1 1 cosPQ 2 2 2cosPQ

    By coordinate geometry

  • Sum & Difference of Anglesy

    x

    1

    P

    1

    x

    y

    cos1x

    cosx sin

    1sin

    y

    y

    cos ,sin

    Q cos ,sin

    By trigonometry

    2 2 21 1 2 1 1 cosPQ 2 2 2cosPQ

    2 22 cos cos sin sinPQ By coordinate geometry

  • Sum & Difference of Anglesy

    x

    1

    P

    1

    x

    y

    cos1x

    cosx sin

    1sin

    y

    y

    cos ,sin

    Q cos ,sin

    By trigonometry

    2 2 21 1 2 1 1 cosPQ 2 2 2cosPQ

    2 22 cos cos sin sinPQ 2 2 2 2 2cos 2cos cos cos sin 2sin sin sinPQ

    By coordinate geometry

  • Sum & Difference of Anglesy

    x

    1

    P

    1

    x

    y

    cos1x

    cosx sin

    1sin

    y

    y

    cos ,sin

    Q cos ,sin

    By trigonometry

    2 2 21 1 2 1 1 cosPQ 2 2 2cosPQ

    2 22 cos cos sin sinPQ 2 2 2 2 2cos 2cos cos cos sin 2sin sin sinPQ

    By coordinate geometry

    2 2 2cos cos 2sin sinPQ

  • 2 2cos 2 2cos cos 2sin sin

  • 2 2cos 2 2cos cos 2sin sin cos cos cos sin sin

  • 2 2cos 2 2cos cos 2sin sin cos cos cos sin sin

    Replace with

  • 2 2cos 2 2cos cos 2sin sin cos cos cos sin sin

    Replace with cos cos cos sin sin

  • 2 2cos 2 2cos cos 2sin sin cos cos cos sin sin

    Replace with cos cos cos sin sin cos cos (even function i.e. )f x f x

  • 2 2cos 2 2cos cos 2sin sin cos cos cos sin sin

    Replace with cos cos cos sin sin cos cos (even function i.e. )f x f x sin sin (odd function i.e. )f x f x

  • 2 2cos 2 2cos cos 2sin sin cos cos cos sin sin

    Replace with cos cos cos sin sin cos cos (even function i.e. )f x f x sin sin (odd function i.e. )f x f x

    cos cos cos sin sin

  • 2 2cos 2 2cos cos 2sin sin cos cos cos sin sin

    Replace with cos cos cos sin sin cos cos (even function i.e. )f x f x sin sin (odd function i.e. )f x f x

    cos cos cos sin sin

    sinsincoscoscos

  • 2 2cos 2 2cos cos 2sin sin cos cos cos sin sin

    Replace with cos cos cos sin sin cos cos (even function i.e. )f x f x sin sin (odd function i.e. )f x f x

    cos cos cos sin sin cos, cos, sin, sin

    sinsincoscoscos

  • 2 2cos 2 2cos cos 2sin sin cos cos cos sin sin

    Replace with cos cos cos sin sin cos cos (even function i.e. )f x f x sin sin (odd function i.e. )f x f x

    cos cos cos sin sin cos, cos, sin, sin

    sinsincoscoscos

  • 2 2cos 2 2cos cos 2sin sin cos cos cos sin sin

    Replace with cos cos cos sin sin cos cos (even function i.e. )f x f x sin sin (odd function i.e. )f x f x

    cos cos cos sin sin cos, cos, sin, sin

    sinsincoscoscos

  • 2 2cos 2 2cos cos 2sin sin cos cos cos sin sin

    Replace with cos cos cos sin sin cos cos (even function i.e. )f x f x sin sin (odd function i.e. )f x f x

    cos cos cos sin sin cos, cos, sin, sin

    sinsincoscoscos

  • 2 2cos 2 2cos cos 2sin sin cos cos cos sin sin

    Replace with cos cos cos sin sin cos cos (even function i.e. )f x f x sin sin (odd function i.e. )f x f x

    cos cos cos sin sin cos, cos, sin, sin

    sinsincoscoscos

  • 2 2cos 2 2cos cos 2sin sin cos cos cos sin sin

    Replace with cos cos cos sin sin cos cos (even function i.e. )f x f x sin sin (odd function i.e. )f x f x

    cos cos cos sin sin cos, cos, sin, sin

    If its not the sine

    sinsincoscoscos

  • 2 2cos 2 2cos cos 2sin sin cos cos cos sin sin

    Replace with cos cos cos sin sin cos cos (even function i.e. )f x f x sin sin (odd function i.e. )f x f x

    cos cos cos sin sin cos, cos, sin, sin

    If its not the sine,its not the sign

    sinsincoscoscos

  • Replace with 90

  • Replace with 90 cos 90 cos 90 cos sin 90 sin

  • Replace with 90 cos 90 cos 90 cos sin 90 sin

    cos 90 cos 90 cos sin 90 sin

  • Replace with 90 cos 90 cos 90 cos sin 90 sin

    cos 90 cos 90 cos sin 90 sin sin sin cos cos sin

  • Replace with 90 cos 90 cos 90 cos sin 90 sin

    cos 90 cos 90 cos sin 90 sin sin sin cos cos sin

    Replace with

  • Replace with 90 cos 90 cos 90 cos sin 90 sin

    cos 90 cos 90 cos sin 90 sin sin sin cos cos sin

    Replace with sin sin cos cos sin

  • Replace with 90 cos 90 cos 90 cos sin 90 sin

    cos 90 cos 90 cos sin 90 sin sin sin cos cos sin

    Replace with sin sin cos cos sin sin sin cos cos sin

  • Replace with 90 cos 90 cos 90 cos sin 90 sin

    cos 90 cos 90 cos sin 90 sin sin sin cos cos sin

    Replace with sin sin cos cos sin sin sin cos cos sin

    sincoscossinsin

  • Replace with 90 cos 90 cos 90 cos sin 90 sin

    cos 90 cos 90 cos sin 90 sin sin sin cos cos sin

    Replace with sin sin cos cos sin sin sin cos cos sin

    sincoscossinsin sin, cos, cos, sin

  • Replace with 90 cos 90 cos 90 cos sin 90 sin

    cos 90 cos 90 cos sin 90 sin sin sin cos cos sin

    Replace with sin sin cos cos sin sin sin cos cos sin

    sincoscossinsin sin, cos, cos, sin

  • Replace with 90 cos 90 cos 90 cos sin 90 sin

    cos 90 cos 90 cos sin 90 sin sin sin cos cos sin

    Replace with sin sin cos cos sin sin sin cos cos sin

    sincoscossinsin sin, cos, cos, sin

  • Replace with 90 cos 90 cos 90 cos sin 90 sin

    cos 90 cos 90 cos sin 90 sin sin sin cos cos sin

    Replace with sin sin cos cos sin sin sin cos cos sin

    sincoscossinsin sin, cos, cos, sin

  • Replace with 90 cos 90 cos 90 cos sin 90 sin

    cos 90 cos 90 cos sin 90 sin sin sin cos cos sin

    Replace with sin sin cos cos sin sin sin cos cos sin

    sincoscossinsin sin, cos, cos, sin

  • Replace with 90 cos 90 cos 90 cos sin 90 sin

    cos 90 cos 90 cos sin 90 sin sin sin cos cos sin

    Replace with sin sin cos cos sin sin sin cos cos sin

    sincoscossinsin sin, cos, cos, sin

    If its the sine

  • Replace with 90 cos 90 cos 90 cos sin 90 sin

    cos 90 cos 90 cos sin 90 sin sin sin cos cos sin

    Replace with sin sin cos cos sin sin sin cos cos sin

    sincoscossinsin sin, cos, cos, sin

    If its the sine,its the sign

  • tan

  • tan

    sintan

    cos

  • tan

    sintan

    cos

    sin cos cos sintancos cos sin sin

  • tan

    sintan

    cos

    sin cos cos sintancos cos sin sin

    sin cos cos sincos cos cos costan cos cos sin sincos cos cos cos

  • tan

    sintan

    cos

    sin cos cos sintancos cos sin sin

    sin cos cos sincos cos cos costan cos cos sin sincos cos cos cos

    tan tantan1 tan tan

  • tan

    sintan

    cos

    sin cos cos sintancos cos sin sin

    sin cos cos sincos cos cos costan cos cos sin sincos cos cos cos

    tan tantan1 tan tan

    Replace with

  • tan

    sintan

    cos

    sin cos cos sintancos cos sin sin

    sin cos cos sincos cos cos costan cos cos sin sincos cos cos cos

    tan tantan1 tan tan

    Replace with

    tan tantan

    1 tan tan

  • tan

    sintan

    cos

    sin cos cos sintancos cos sin sin

    sin cos cos sincos cos cos costan cos cos sin sincos cos cos cos

    tan tantan1 tan tan

    Replace with

    tan tantan

    1 tan tan tan tan

    (odd function i.e. )f x f x

  • tan

    sintan

    cos

    sin cos cos sintancos cos sin sin

    sin cos cos sincos cos cos costan cos cos sin sincos cos cos cos

    tan tantan1 tan tan

    Replace with

    tan tantan

    1 tan tan tan tan

    (odd function i.e. )f x f x tan tantan1 tan tan

  • tantan1tantantan

  • tan plus tan

    tantan1tantantan

  • tan plus tan

    on one minus tan, tan

    tantan1tantantan

  • tan plus tan

    on one minus tan, tan

    tantan1tantantan

    Sum and Difference of Angles

  • tan plus tan

    on one minus tan, tan

    tantan1tantantan

    Sum and Difference of Angles

    sincoscossinsin

  • tan plus tan

    on one minus tan, tan

    tantan1tantantan

    Sum and Difference of Angles

    sincoscossinsin sinsincoscoscos

  • tan plus tan

    on one minus tan, tan

    tantan1tantantan

    Sum and Difference of Angles

    sincoscossinsin sinsincoscoscos

    tantan1tantantan

  • e.g. i Expand cos 2 3

  • e.g. i Expand cos 2 3 cos 2 3 cos 2 cos3 sin 2 sin 3

  • e.g. i Expand cos 2 3 cos 2 3 cos 2 cos3 sin 2 sin 3

    tan 20 tan10(ii) Simplify 1 tan 20 tan10

  • e.g. i Expand cos 2 3 cos 2 3 cos 2 cos3 sin 2 sin 3

    tan 20 tan10(ii) Simplify 1 tan 20 tan10

    tan 20 tan10 tan 20 101 tan 20 tan10

  • e.g. i Expand cos 2 3 cos 2 3 cos 2 cos3 sin 2 sin 3

    tan 20 tan10(ii) Simplify 1 tan 20 tan10

    tan 20 tan10 tan 20 101 tan 20 tan10

    tan30

  • e.g. i Expand cos 2 3 cos 2 3 cos 2 cos3 sin 2 sin 3

    tan 20 tan10(ii) Simplify 1 tan 20 tan10

    tan 20 tan10 tan 20 101 tan 20 tan10

    tan30 13

  • e.g. i Expand cos 2 3 cos 2 3 cos 2 cos3 sin 2 sin 3

    tan 20 tan10(ii) Simplify 1 tan 20 tan10

    tan 20 tan10 tan 20 101 tan 20 tan10

    tan30 13

    (iii) Find the exact value of sin15

  • e.g. i Expand cos 2 3 cos 2 3 cos 2 cos3 sin 2 sin 3

    tan 20 tan10(ii) Simplify 1 tan 20 tan10

    tan 20 tan10 tan 20 101 tan 20 tan10

    tan30 13

    (iii) Find the exact value of sin15

    sin15 sin 45 30

  • e.g. i Expand cos 2 3 cos 2 3 cos 2 cos3 sin 2 sin 3

    tan 20 tan10(ii) Simplify 1 tan 20 tan10

    tan 20 tan10 tan 20 101 tan 20 tan10

    tan30 13

    (iii) Find the exact value of sin15

    sin15 sin 45 30 sin 45 cos30 cos 45 sin30

  • e.g. i Expand cos 2 3 cos 2 3 cos 2 cos3 sin 2 sin 3

    tan 20 tan10(ii) Simplify 1 tan 20 tan10

    tan 20 tan10 tan 20 101 tan 20 tan10

    tan30 13

    (iii) Find the exact value of sin15

    sin15 sin 45 30 sin 45 cos30 cos 45 sin30

    1 3 1 12 22 2

  • e.g. i Expand cos 2 3 cos 2 3 cos 2 cos3 sin 2 sin 3

    tan 20 tan10(ii) Simplify 1 tan 20 tan10

    tan 20 tan10 tan 20 101 tan 20 tan10

    tan30 13

    (iii) Find the exact value of sin15

    sin15 sin 45 30 sin 45 cos30 cos 45 sin30

    1 3 1 12 22 2

    3 1

    2 2

  • 2 1iv If sin and cos , find sin3 4

  • 2 1iv If sin and cos , find sin3 4

    23

    5

  • 2 1iv If sin and cos , find sin3 4

    23

    5

    1

    4 15

  • 2 1iv If sin and cos , find sin3 4

    23

    5

    1

    4 15

    sin sin cos cos sin

  • 2 1iv If sin and cos , find sin3 4

    23

    5

    1

    4 15

    sin sin cos cos sin 2 1 5 153 4 3 4

  • 2 1iv If sin and cos , find sin3 4

    23

    5

    1

    4 15

    sin sin cos cos sin 2 1 5 153 4 3 4

    2 5 3

    12

  • 2 1iv If sin and cos , find sin3 4

    23

    5

    1

    4 15

    sin sin cos cos sin 2 1 5 153 4 3 4

    2 5 3

    12

    Exercise 14D; 1ade, 2bce, 5ac, 7, 9ac, 10ac, 12, 13ac, 16ab, 17, 23

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