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Problems on Submultiple Angles

We will learn how to solve the problems on submultiple angles formula.

1. If sin x = 3/5 and 0 < x < π2, find the value of tan x2

Solution:

tan x2

= 1cosx1+cosx

= 1451+45

= 19

= 13

2. Show that, (sin2 24° - sin2 6° ) (sin2 42° - sin2 12°) = 116

Solution:  

L.H.S. = 1/4 (2 sin2 24˚ - 2 sin2 6˚)(2 sin2 42˚ - 2 sin2 12˚)

= ¼ [(1- cos 48°) - (1 - cos 12°)] [(1 - cos 84° ) - (1 - cos 24°)]

= ¼ (cos 12° - cos 48°)(cos 24° - cos 84°)

= ¼ (2 sin 30° sin 18° ) (2 sin 54° sin 30°)

= ¼ [2 ∙ ½ ∙ sin 18°] [2 ∙ sin(90° - 36°) × ½]

= ¼ sin 18° ∙ cos 36°

= 145145+14

= 14 × 416

= 116 = R.H.S.              Proved.

 

3. If tan x = ¾ and x lies in the third quadrant, find the values of sin x2, cos x2  and tan x2.

Solution:

As x lies in the third quadrant, cos x is negative

sec2 x = 1 + tan2 x = 1 + (3/4)2 = 1 + 916 = 2516

⇒ cos2 x = 2516

⇒ cos x = ± 45, but cos x is negative

Therefore, cos x = -45

Also π < x < 3π2

π2 < x2 < 3π4

x2  lies in second quadrant

⇒ cos x2 is –ve and sin x2 is +ve.

Therefore, cos x2 = -1+cosx2 = -1452 = - 110

sin x2 = -1cosx2 = 1(45)2 = 910 =  310

tan x2 = sinx2cosx2 = 310(101) = -3

 

4. Show that using the formula of submultiple angles tan 6˚ tan 42˚ tan 66˚ tan 78˚ = 1.

Solution:  

L.H.S = tan 6˚ tan 42˚ tan 66˚ tan 78˚

= (2sin6˚sin66˚)(2sin42˚sin78˚)(2cos6˚cos66˚)(2cos42˚cos78˚)

= (cos60˚cos72˚)(cos36˚cos120˚)(cos60˚+cos72˚)(cos36˚+cos120˚)

= (12sin18˚)(cos36˚+12)(12+sin18˚)(cos36˚12)[Since, cos 72˚ = cos (90˚ - 18˚) = sin 18˚ and cos 120˚ = cos ( 180˚ - 60˚) = - cos 60˚ = -1/2]

= (12514)(5+14+12)(12+514)(5+1412), [ putting the values of sin 18˚ and cos 36˚]

= (35)(3+5)(5+1)(51)

= 9551

= 44

= 1 = R.H.S.              Proved.

 

5.  Without using table prove that, sin 12° sin 48° sin 54˚ = 18

Solution:

L. H. S. = sin 12° sin 48° sin 54° 

= 12 (2 sin 12°sin 48°) sin (90°- 36°) 

= 12 [cos 36°- cos 60°] cos 36°

= 12 [√5+14 - 12] 5+14, [Since, cos 36˚ = 5+14]

= 125145+14

= 432

= 18 =  R.H.S.              Proved.

 Submultiple Angles





11 and 12 Grade Math

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