1 press ctrl-a ©g dear 2010 – not to be sold/free to use sum & differences of angles stage 6...
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Press Ctrl-APress Ctrl-A©G Dear 2010 – Not to be sold/Free to use
Sum & Differences Sum & Differences of anglesof angles
Stage 6 - Year 11Stage 6 - Year 11
Mathematic Extension 1(Preliminary)
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We find the sum or difference of anglesto make the hard questions easier.
End ofSlide
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1
1-1x
yx-y
(cos x, sin x) B
A (cos y, sin y)
x - y
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AB2 = (cos x – cos y)2 + (sin x – sin y)2
End ofSlide
cos(x – y) = cos x cos y + sin x sin yProof part 1 - distance formula
d2 = (x2–x1)2 + (y2–y1)2
+ (cos2y + sin2y)- 2cos x cos y- 2sin x sin y
= 2- 2(cos x cos y + sin x sin y)
1
Xo-yo
= cos2x – 2 cos x cos y + cos2y
+ sin2x – 2 sin x sin y + sin2y
= (cos2x + sin2x)
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End ofSlide
cos(x – y) = cos x cos y + sin x sin yProof part 2 – cosine rule
a2 = b2 + c2 – 2bc cos AAB2 = 12 + 12 – 2(1)(1) cos(x - y)
AB2 = 2 – 2 cos(x - y)2
Xo-yo
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End ofSlide
cos(x – y) = cos x cos y + sin x sin yProof part 3 – from 1 and 2
2 – 2 cos(x-y) =2
2 - 2(cos x cos y + sin x sin y)1
x x
xo-yo
–2 cos(x-y) =2
-2(cos x cos y + sin x sin y)1
x xcos(x-y) =cos x cos y + sin x sin y
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End ofSlide
-1
1
1-1x yx+y
(cos x, sin x) B A (cos y, sin y)
1 1
xo + yo
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End ofSlide
cos(x + y) = cos x cos y - sin x sin y
Replace y with -y
cos(x–y) = cos x cos y + sin x sin y
cos(x–(-y)) = cos x cos(-y) + sin x sin(-y)
cos(x+y) = cos x cos y + sin x (-sin y)
cos(x+y) = cos x cos y - sin x sin y
xo+yo
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End ofSlide
-1
sin(x + y) = sin x cos y + cos x sin y
Replace x with 90o-x
cos(x–y) = cos x cos y + sin x sin y
cos((90o-x)– y) = cos(90o-x) cos y + sin(90o-x) sin y
cos(90o-(x+y)) = sin x cos y + cos x sin y
sin(x+y) = sin x cos y + cos x sin y
xo+yo
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End ofSlide
sin(x - y) = sin x cos y - cos x sin y
Replace y with -ysin (x+y) = sin x cos y + cos x
sin ysin (x+(-y)) = sin x cos (-y) + cos x sin
(-y)sin (x-y) = sin x cos y + cos x (-sin y)
sin(x-y) = sin x cos y - cos x sin y
xo-yo
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End ofSlide
sin(x + y) = sin x cos y + cos x sin y
Replace x with 90o-x
cos(x – y) = cos x cos y + sin x sin y
cos((90o-x) – y) = cos (90o-x) cos y + sin (90o-x) sin y
cos(90o- (x+y)) = sin x cos y + cos x sin y
sin(x+y) = sin x cos y + cos x sin y
xo+yo
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=
sin x cos y + cos x sin ycos x cos y
End ofSlide
tan x + tan y1 – tan x tan y
tan (x+y) =
xo+yo
tan(x+y) =sin(x+y)cos(x+y) cos x cos y - sin x sin y=sin x cos y + cos x sin y
cos x cos y + sin x sin ycos x cos y
=sin x cos ycos x cos y
+cos x sin ycos x cos y
cos x cos ycos x cos y
-sin x sin ycos x cos y
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xxxx xxxxxxxx xxxx
+ tan y= tan x1- tan x
1
tan y
tan(x+y) =
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=
sin x cos y - cos x sin ycos x cos y
End ofSlide
tan x - tan y1 – tan x tan y
tan (x-y) =
xo-yo
tan(x-y) = sin(x-y)cos(x-y) cos x cos y + sin x sin y=sin x cos y - cos x sin y
cos x cos y + sin x sin ycos x cos y
=sin x cos ycos x cos y
-cos x sin ycos x cos y
cos x cos ycos x cos y
+sin x sin ycos x cos y
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xxxxxxxx
xxxx xxxxxxxx xxxx
- tan y= tan x1+ tan x
1
tan y
tan(x-y) =
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End ofSlide
cos(x – y) = cos x cos y + sin x sin ycos(x + y) = cos x cos y - sin x sin y
sin(x + y) = sin x cos y + cos x sin y
sin(x - y) = sin x cos y - cos x sin ytan x + tan y1 – tan x tan y
tan (x+y) =
tan x - tan y1 – tan x tan y
tan (x-y) =