Exercise 10.4 (Solutions)
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Textbook of Algebra and Trigonometry for Class XI
Merging man and maths
Available online @ http://www.mathcity.org, Version: 2.0.0
Page 336
Question # 1
(i) Since 2sin α cos β = sin(α + β ) + sin(α − β )
Put α = 3θ and β = θ
2sin 3θ cosθ = sin(3θ + θ ) + sin(3θ − θ )
= sin 4θ + sin 2θ
(ii) Since 2cosα cos β = cos (α + β ) + cos (α − β )
Put α = 5θ and β = 3θ
2cos5θ cos3θ = cos(5θ + 3θ ) − cos(5θ − 3θ )
= cos8θ − cos 2θ
(iii) Since 2sin α cos β = sin(α + β ) + sin(α − β )
Put α = 5θ and β = 2θ
2sin 5θ cos 2θ = sin(5θ + 2θ ) + sin(5θ − 2θ )
= sin 7θ + sin 3θ
1
⇒ sin 5θ cos 2θ = ( sin 7θ + sin 3θ )
2
Since − 2sin α sin β = cos(α + β ) − cos(α − β )
Put α = 7θ and β = 2θ
− 2sin α sin β = cos(α + β ) − cos(α − β )
−2sin 7θ sin 2θ = cos(7θ + 2θ ) − cos(7θ − 2θ )
= cos9θ − cos5θ
2sin 7θ sin 2θ = cos5θ − cos9θ
(iv)
(v)
Since 2cosα sin β = sin(α + β ) − sin(α − β )
Put α = x + y , β = x − y
2cos( x + y )sin( x − y ) = sin( x + y + x − y ) − sin( x + y − x + y )
= sin 2 x − sin 2 y
1
⇒ cos( x + y )sin( x − y ) = ( sin 2 x − sin 2 y )
2
Since 2cosα cos β = cos (α + β ) + cos (α − β )
(vi)
Put α = 2 x + 30 and β = 2 x − 30
(
) (
)
(
)
(
2cos 2 x + 30 cos 2 x − 30 = cos 2 x + 30 + 2 x − 30 + cos 2 x + 30 − 2 x + 30
( )
1
cos ( 2 x + 30 ) cos ( 2 x − 30 ) = ( cos 4 x + cos 60 )
2
= cos ( 4 x ) + cos 60
⇒
(vi)
Since
− 2sin α sin β = cos(α + β ) − cos(α − β )
)
FSc-I / Ex 10.4 - 2
α = 12 and β = 46
Put
− 2sin12 sin 46 = cos(12 + 46) − cos(12 − 46)
= cos58 − cos(−34)
= cos58 − cos34
∵ cos(−θ ) = cosθ
1
⇒ sin12 sin 46 = − ( cos58 − cos34 )
2
1
= ( cos34 − cos58 )
2
Since − 2sin α sin β = cos(α + β ) − cos(α − β )
(viii)
Put α = x + 45 and β = x − 45
(
) (
)
{
}
{
}
− 2sin x + 45 sin x − 45 = cos ( x + 45 ) + ( x − 45 ) − cos ( x + 45 ) − ( x − 45 )
= cos 2 x − cos90
1
⇒ sin x + 45 sin x − 45 = cos90 − cos 2 x
2
(
) (
)
Question # 2
(i)
α + β 
α − β 
Since sin α + sin β = 2sin 
 cos 

 2 
 2 
Put α = 5θ , β = 3θ
 5θ + 3θ 
 5θ − 3θ 
sin 5θ + sin 3θ = 2sin 
 cos 

 2 
 2 
 8θ 
 2θ 
= 2sin   cos   = 2sin 4θ cosθ
 2 
 2 
α + β  α − β 
Since sin α − sin β = 2cos 
 sin 

2

  2 
Put α = 8θ , β = 4θ
 8θ + 4θ   8θ − 4θ 
sin8θ − sin 4θ = 2cos 
 sin 

 2   2 
= 2cos 6θ sin 2θ
(iii) Do yourself ☺
(ii)
(iv)
(v)
Since
Since
α + β  α − β 
cosα − cos β = −2sin 
 sin 

 2   2 
Put α = 7θ , β = θ
 7θ + θ   7θ − θ 
 8θ
cos 7θ − cosθ = −2sin 
 sin 
 = −2sin 
 2   2 
 2
= −2sin 4θ sin 3θ
α + β
cosα + cos β = 2cos 
 2

α − β 
 cos 


 2 
  6θ 
 sin  
  2 
FSc-I / Ex 10.4 - 3
Put α = 12 , β = 48
 12 + 48   12 − 48 
cos12 + cos 48 = 2cos 
 cos 

 2   2 
 60 
 −36 
= 2cos   cos 
 = 2cos30cos ( −18 )
 2 
 2 
= 2cos30 cos18
∵ cos(−θ ) = cosθ
(vi)
α + β 
α − β 
sin α + sin β = 2sin 
 cos 

 2 
 2 
Put α = x + 30 , β = x − 30
 x + 30 + x − 30 
 x + 30 − x + 30 
sin( x + 30) + sin( x − 30) = 2sin 
 cos 

2
2




 2x 
 60 
= 2sin   cos   = 2sin x cos30
 2 
 2 
Since
Question # 3
sin 3 x − sin x
cos x − cos3 x
 3x + x   3x − x 
 4x   2x 
2cos 
cos   sin  
 sin 

 2   2  =
 2   2 
=
 x + 3x   x − 3x 
 4 x   −2 x 
−2sin 
− sin   sin 
 sin 


 2   2 
 2   2 
cos ( 2 x ) sin ( x )
=
= cot 2 x = R.H.S
+ sin ( 2 x ) sin ( x )
(i) L.H.S =
(ii) Do yourself ☺
sin α − sin β
(iii) L.H.S =
sin α + sin β
α + β  α − β
2cos 
 sin 
2   2

=
α + β  α − β
2sin 
 cos 
 2   2


 = cot  α + β


 2


 α − β 
 tan 
 = R.H.S
  2 
Question # 4
(i) L.H.S = cos 20 + cos100 + cos140
= cos100 + cos 20 + cos140
(
)
 100 + 20   100 − 20 
= 2cos 
 cos 
 + cos140
2
2

 

1
= 2cos60 cos 40 + cos140 = 2   cos 40 + cos140
2
FSc-I / Ex 10.4 - 4
 140 + 40 
 140 − 40 
= cos140 + cos 40 = 2cos 
 cos 

2
2




= 2cos90 cos50 = 2 ( 0 ) cos50 = 0 = R.H.S
(ii)
π
 π

L.H.S = sin  − θ  sin  + θ 
4
 4

π
π
 π
 π

=  sin cosθ − cos sin θ  sin cosθ + cos sin θ 
4
4
4
4



1
1
 1
 1

=
cosθ −
sin θ 
cosθ +
sin θ 
2
2
 2
 2

2
2
1
1
 1
  1

=
cosθ  − 
sin θ  = cos 2 θ − sin 2 θ
2
2
 2
  2

1
1
= cos 2 θ − sin 2 θ = cos 2θ = R.H.S
2
2
sin θ + sin 3θ + sin 5θ + sin 7θ
(iii) L.H.S =
cosθ + cos3θ + cos5θ + cos 7θ
( sin 7θ + sinθ ) + ( sin 5θ + sin 3θ )
=
( cos 7θ + cosθ ) + ( cos5θ + cos3θ )
(
)
 7θ + θ 
 7θ − θ 
 5θ + 3θ 
 5θ − 3θ 
2sin 
 cos 
 + 2sin 
 cos 

 2 
 2 
 2 
 2 
=
 7θ + θ 
 7θ − θ 
 5θ + 3θ 
 5θ − 3θ 
2cos 
 cos 
 + 2cos 
 cos 

 2 
 2 
 2 
 2 
2sin 4θ cos3θ + 2sin 4θ cosθ
=
2cos 4θ cos3θ + 2cos 4θ cosθ
2sin 4θ ( cos3θ + cosθ )
sin 4θ
=
=
= tan 4θ = R.H.S
2cos 4θ ( cos3θ + cosθ )
cos 4θ
Question # 5
(i)
L.H.S = cos 20 cos 40 cos 60 cos80
1
1
= cos 20 cos 40   cos80 = cos80 cos 40 cos 20
2
2
1
1
= 2cos80 cos 40 cos 20 = ( cos ( 80 + 40 ) + cos ( 80 − 40 ) ) cos 20
4
4
1
1 1

= cos120 + cos 40 cos 20 =  − + cos 40  cos 20
4
4 2

1
1
1 1
= − cos 20 + cos 40 cos 20 = − + 2cos 40 cos 20
8
4
8 8
1
1
= − cos 20 + ( cos(40 + 20) + cos(40 − 20) )
8
8
(
(
)
)
(
)
FSc-I / Ex 10.4 - 5
1
1
1
1 1

= − cos 20 + ( cos 60 + cos 20 ) = − cos 20 +  + cos 20 
8
8 2
8
8

1
1 1
1
= − cos 20 + + cos 20 =
= R.H.S
8
16 8
16
(ii)
2π
π
4π
sin sin
9
9
3
9
180
2(180 )
(180 )
4(180 )
= sin
sin
sin
sin
9
9
3
9
L.H.S = sin
π
sin
= sin 20 sin 40 sin 60 sin 80 = sin 20 sin 40
∵ π = 180
3
sin 80
2
3
3
sin 80 sin 40 sin 20 = −
−2sin 80 sin 40 ) sin 20
(
2
4
3
=−
( cos(80 + 40) − cos(80 − 40) ) sin 20
4
3
3 1

=−
cos120 − cos 40 ) sin 20 = −
(
 − − cos 40  sin 20
4
4  2

=
3
3
3
3
sin 20 +
cos 40 sin 20 =
sin 20 +
2cos 40 sin 20 )
(
8
4
8
8
3
3
=
sin 20 +
( sin(40 + 20) − sin(40 − 20) )
8
8

3
3
3
3 3
=
sin 20 +
sin 60 − sin 20 ) =
sin 20 +
− sin 20 
(

8
8  2
8
8

=
=
(iii)
3
3
3
3
sin 20 + −
sin 20 =
= R.H.S
8
16 8
16
Do yourself as above ☺
Rana Hamza
Book:
2012-14
Error Analyst
Harvard College of Commerce and Sciences,RWP
Exercise 10.4
Text Book of Algebra and Trigonometry Class XI
Punjab Textbook Board, Lahore.
Made by: Atiq ur Rehman ([email protected])
Available online at http://www.MathCity.org in PDF Format
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Updated: 29-12-2014.