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Additional formulae. sin (A + B) = sin A cos B + sin B cos A. sin (A - B) = sin A cos B - sin B cos A. cos (A + B) = cos A cos B - sin A sin B. cos (A - B) = cos A cos B + sin A sin B. Examples. Find the exact value of sin 75 . - PowerPoint PPT Presentation
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Additional formulaesin (A + B) = sin A cos B + sin B cos Asin (A - B) = sin A cos B - sin B cos Acos (A + B) = cos A cos B - sin A sin Bcos (A - B) = cos A cos B + sin A sin B
1
tan A tan Btan( A B )
tan Atan B
1
tan A tan Btan( A B )
tan Atan B
ExamplesFind the exact value of sin 75
sin (A + B) = sin A cos B + sin B cos Asin (30 + 45) = sin 30 cos 45 + sin 45 cos 30
1 2 2 3
2 2 2 2
2 6 2 6
4 4 4
ExamplesExpress cos (x + /3) in terms of cos x and sin x
1 3
2 2cos x sin x
cos (A + B) = cos A cos B - sin A sin B
cos (x + /3) = cos x cos /3 - sin /3 sin x
Examplessin( A B )
Pr ove that tan A tan Bcos Acos B
L.H.S.sin A sin B
tan A tan Bcos A cos B
sin Acos B sin B cos A
cos Acos B
sin( A B )
cos Acos B
=
R.H.S.
Double angle formulaesin (A + B) = sin A cos B + sin B cos Asin (A + A) = sin A cos A + sin A cos A sin 2A = 2 sin A cos A
cos (A + B) = cos A cos B - sin A sin Bcos (A + A) = cos A cos A- sin A sin Acos (A + A) = cos2A - sin2A
cos 2A = cos2A - sin2A
cos 2A = 2cos2A - 1
cos 2A = 1 – 2sin2A
Double angle formulae
1
tan A tan Btan( A B )
tan Atan B
1
tan A tan Atan( A A )
tan Atan A
2
22
1
tan Atan A
tan A
2 2Pr ovethat tan A tan A tan Asec A 2 2
2 2 2
2 2 1 1
1 1 1
tan A tan A tan A( tan A ) tan A( tan A )tan A
tan A tan A tan A
2
22
1 2
tan Asec A tan Atan Asec A
tan A cos A
ExamplesGiven that cos A = 2/3, find the exact value of cos 2A.
cos 2A = 2cos2A - 12
22 1
3
8 11
9 9
Given that sin A = ¼ , find the exact value of sin 2A.
sin 2A = 2 sin A cos A
1 15 152
4 4 8 A
4 1
15
Solving equationsSolve cos 2A + 3 + 4 cos A = 0 for 0 x 2
=2 cos2A - 1+ 3 + 4 cos A = 0
=2 cos2A + 4 cos A + 2= 0
= cos2A + 2 cos A + 1 = 0
= cos2A + 2 cos A + 1 = 0
= (cos A + 1)2 = 0
= cos A = - 1
A =
Solving equationsSolve sin 2A = sin A for - x
=2sin A cos A = sin A
=2 sin A cos A – sin A = 0
= sin A(2 cos A – 1) = 0
sin A = 0 or cos A = ½
sin A = 0 A = - or 0 or
cos A = ½ A = - /3 or /3
Complete solution: A = - or - /3 or 0 or /3 or
Solving equationsSolve tan 2A + 5 tan A = 0 for 0 x 2
Complete solution: A= 0.97 , 2.27, 4.01, 5.41c 0, or 2
22
25 2 5 1 0
1
tan Atan A tan A tan A( tan A )
tan A
22 5 1 0tan A[ ( tan A )]
22 5 5 0tan A[ tan A )
27 5 0tan A[ tan A )
tan A = 0 A = 0 or or 2
7 – 5tan2 A = 0 tan A = 7/5 A = 0.97 , 2.27, 4.01 or 5.41c
Harmonic form
If a and b are positive a sin x + b cos x can be written in the form R sin( x + )
a cos x + b sin x can be written in the form R cos( x - )
a sin x - b cos x can be written in the form R sin( x - )
a cos x - b sin x can be written in the form R cos( x + )
2 2R a b
R cos a and R sin b
ExamplesExpress 3 cos x + 4 sin x in the form R cos( x - )
R cos( x - ) = R cos x cos + R sin x sin
3 cos x + 4 sin x = R cos x cos + R sin x sin
R cos = 3 [1] R sin = 4 [2]
[1]2 + [2]2 : R2 sin2 x + R2 cos2 x = 32 + 42
R2(sin2 x + cos2 x ) = 32 + 42
R2= 32 + 42 = 25 R = 5
[2] [1]: tan = 4/3 = 53.1
3 cos x + 4 sin x = 5 cos( x + 53.1 )
ExamplesExpress 12 cos x + 5 sin x in the form R sin( x + )
R sin( x + ) = R sin x cos + R cos x sin
12 cos x + 5 sin x = R sin x cos + R cos x sin
R cos = 12 [1] R sin = 5 [2]
[1]2 + [2]2 : R2 cos2 x + R2 sin2 x = 122 + 52
R2(cos2 x + sin2 x ) = 122 + 52
R2= 122 + 52 = 169 R = 13
[2] [1]: tan = 5/12 = 22.6
12 cos x + 5 sin x = 13 sin( x + 22.6 )
ExamplesExpress cos x - 3 sin x in the form R cos( x + )
R cos( x + ) = R cos x cos - R sin x sin
cos x - 3 sin x = R cos x cos - R sin x sin
R cos = 1 [1] R sin = 3 [2]
[1]2 + [2]2 : R2 cos2 x + R2 sin2 x = 12 + (3 ) 2
R2(cos2 x + sin2 x ) = 12 + 3
R2= 1 + 3 = 4 R = 2
[2] [1]: tan = 3 = 60
cos x + 3 sin x = 2 cos( x + 60 )
Solving equationsSolve 7 sin x + 3 cos x = 6 for 0 x 2
R sin( x + ) = R sin x cos + R cos x sin
7 sin x + 3 cos x = R sin x cos + R cos x sin
R cos = 7 [1] R sin = 3 [2]
R2 = 72 + 32 R = 7.62
[2] [1]: tan = 3/7 = 0.405c (Radians)
7 sin x + 3 cos x = 7.62 sin( x + 0.405)
7.62 sin( x + 0.405 ) = 6 x + 0.405 = sin-1(6/7.62)
x + 0.405 = 0.907 or 2.235
x = 0.502c or 1.830c