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
sin1
sin
y
y
cos ,sin
Sum & Difference of Anglesy
x
1
P
1
x
y
cos1x
cosx
sin1
sin
y
y
cos ,sin
Q cos ,sin
Sum & Difference of Anglesy
x
1
P
1
x
y
cos1x
cosx
sin1
sin
y
y
cos ,sin
Q cos ,sin
By trigonometry
Sum & Difference of Anglesy
x
1
P
1
x
y
cos1x
cosx
sin1
sin
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
sin1
sin
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
sin1
sin
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
sin1
sin
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
sin1
sin
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
sin1
sin
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 it’s 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 it’s not the sine,it’s 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 it’s 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 it’s the sine,it’s the sign
tan
tan
sin
tancos
tan
sin
tancos
sin cos cos sintancos cos sin sin
tan
sin
tancos
sin cos cos sintancos cos sin sin
sin cos cos sincos cos cos costan cos cos sin sincos cos cos cos
tan
sin
tancos
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
sin
tancos
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
sin
tancos
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
sin
tancos
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
sin
tancos
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 tantan
1 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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