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1)sin 12 X SIN 48 X SIN 54 =____________ 1/16 1/32 1/8 1/4

1)sin 12 X SIN 48 X SIN 54 =____________
  1. 1/16 
  2. 1/32
  3. 1/8
  4. 1/4

Grade:11

1 Answers

Arun
25763 Points
4 years ago
Dear Krisha
 
Let the expression be 'x'.

x = sin(12)sin(48)sin(54) 
   = [sin(12)sin(48)]sin(54) 
   = sin(54) * [cos(48 - 12) - cos(48 + 12)]/2 
   = sin(54) * [cos(36) - cos(60)]/2 
   = sin(54) * [cos(36) - 1/2]/2 
   = cos(36) * [cos(36)/2 - 1/4] 
   = cos²(36)/2 - cos(36)/4. 

cos(36) = (1 + √5)/4.

x = cos²(36)/2 - cos(36)/4 
   = [(1 + √5)/4]²/2 - [(1 + √5)/4]/4 
   = (1 + √5)²/32 - (1 + √5)/16 
   = (6 + 2√5)/32 - (1 + √5)/16 
   = (6 + 2√5)/32 - (2 + 2√5)/32 
   = 4/32 
   = 1/8
 
Regards
Arun (askIITians forum expert)

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