Question

HOW TO FIND THE VALUE OF SIN 18 USING THE TRIGONOMETRIC RATIOS

Answer 1

sin 72° = 2 sin 36° cos 36°                                 by the double angle relationship.
  sin 72° = 4 sin 18° cos 18° (1 - 2sin2 18°)         by the double angle relationship, again.
  cos 18° = 4 sin 18° cos 18° (1 - 2sin2 18°)         by the cofunction properties: sin 72° = cos 18°.
            1 = 4 sin 18° (1 - 2sin2 18°)                       Let x = sin 18°, this is known as 
            1 = 4x(1-2x2)                                             substitution, a useful technique in calculus.
8x3-4x+1 = 0                                                         A product is zero only when one of its factors is zero.
8x3-4x+1 = (2x-1)(4x2+2x-1)=0                           (2x-1)=0 implies x= ½=sin 30° > sin 18° ;
                                                                              Since we know sin is increasing on [0°,90°].
            x = (-2 ±  (4 + 4•4•1))/8                       So we must solve the other factor,
                = (-2 ±  20)/8                                     using the quadratic formula.
                = (-2 ±  4 5)/8
                = (-1 ±  5)/4                                       But the sin 18° > 0, so it cannot be negative.
  sin 18°   = ( (5)-1)/4                                         Hence the middle root is the one we want.