# 11X1 T08 02 sum and difference of angles (2010)

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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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