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4.5 Factorization and Defactorization

Factorization (Sums or Differences into Products)

Defactorization (Products into Sums or Differences)

Note : These identites are valid for both degree and radian measure whenever both sides are defined.

EXAMPLE 1 Verify that sin C + sin D

Solution

We know that sin (a + b) = sin a cos b + cos a
sin b ............(i)

sin (a - b) = sin a cos b - cos a sin b .............(ii)

Adding (i) and (ii) we get,

sin (a + b ) + sin (a - b ) = 2 sin a cos b .............(iii)

substracting (ii) from (i) we get

sin (a + b ) - sin ( a - b ) = 2 cos a cos b ............(iv)

EXAMPLE 2

Verify that sin

Solution

We know that sin (a + b ) = sin a cos b + cos a . sin b ......(i)

and sin ( a - b) = sin a cos b - cos a sin b ......(ii)

Adding (i) and (ii) we get

sin ( a + b) + sin ( a - b ) = 2 sin a . cos b

EXAMPLE 3 Prove that

Solution

EXAMPLE 4 Write cos 7x cos 4x as a sum.

Solution

EXAMPLE 5 Write the difference cos 2 a - cos 8a as a product.

Solution:

EXAMPLE 6 Find the exact value of sin 75 0 + sin 150

Solution :

EXAMPLE 7 Prove that sin 7A + sin 3A = cot 2A

Solution :

EXAMPLE 8 If sin a + sin 3 a = cos a + cos 3 a then prove that, either tan 2a = 1 or cos a = 0

Solution: sin a + sin 3a = cos a + cos 3a ......(Given)

\ 2 sin 2a cos a = 2 cos 2a cos a

\ sin 2a cos a - cos a . cos 2a = 0

\ (sin 2a - cos 2a) cos a = 0

\ either i) sin 2a - cos 2a = 0 or ii) cos a =0

\ i) sin 2a = cos 2a \ tan 2a = 1

ii) cos a = 0

EXAMPLE 9 Find the value of cos 520 + cos 680 + cos 1720

Solution:

EXAMPLE 10 Express as a sum or difference sin 55 0 sin 400

Solution: Using sin a sin B = 1/2 [ cos ( a - b ) - cos ( a + b ) ] we get,

\ sin 550 sin 400 = 1/2 [ cos ( 56 0 - 400) - cos ( 550 + 400)]

= 1/2 [cos 150 - cos 950]

EXAMPLE 11 Solution :

EXAMPLE 12 cos 3x sin2x = 1 /16 (2 cos x - cos 3x - cos 5x)

Solution: cos3x . sin2x = (sin x cos x ) 2cos x

= (1/2 x 2 sin x cos x )2 cos x

= 1/4 (2 sin x cos x )2 cos x

= 1/4 (sin 2x )2 (cos x) ....(\ sin 2x = 2 sin x cos x)

= 1/4 ( sin 2x ) ( sin 2x cos x )

= 1/4 ( sin 2 x ) [1/2 ( sin 3x + sin x)] ....sin a cos b
= 1/2[sin (a +b )cos( a - b )]

= 1/8 [ sin 3x sin 2x + sin 2x sin x ]

= 1/8 [ 1/2 (cos x - cos 5x ) + 1/2 (cos x - cos 3x )]

= 1/16 [ 2 cos x - cos 3x - cos 5x ]

EXAMPLE 13 Simplify cos 3x cos 2x cos x

Solution :

EXAMPLE 14

Solution:

EXAMPLE 15 Find acute angles A and B satisfying

i) cot ( A + B) = 1 , cosec (A - B) = 2 ii) sec A cot B - sec A - 2 cot B + 2 = 0

Solution:

i) \ cot ( A+ B) = 1

\ tan ( A+ B) = 1

\ (A + B) = 450 and

\ cosec (A - B) = 2

\ sin ( A - B) = 1/2
\ A - B = 300

Thus we have A + B = 450 and A - B = 300 , adding we have 2A= 750

A = 37.50and subtracting we get 2B = 150

\ B = 7.50

ii) sec A cot B - sec A - 2 cot B + 2 = 0

\ sec A ( cot B - 1 ) - 2 ( cot B - 1 ) = 0

\ ( sec A - 2 ) ( cot B - 1 ) = 0

\ sec A = 2 or cot B = 1

\ cos A = 1/2 or tan B = 1

\ A = 600 or B = 450

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Index

Trignometric Identities

4.1 Fundamental Identities
4.2 The addition formulas
4.3 The multiple-angle (Double & Half angle) formulas
4.4 Tangent Identities
4.5 Factorization & Defactorization

Supplementary Problems


Chapter 5

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