find value of Sin 18

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 ± [square root of] (4 + 4•4•1))/8 So we must solve the other factor, = (-2 ± [square root of] 20)/8 using the quadratic formula. = (-2 ± [square root of] 4[square root of] 5)/8 = (-1 ± [square root of] 5)/4 But the sin 18° > 0, so it cannot be negative. sin 18° = ([square root of] (5)-1)/4 Hence the middle root is the one we want. read less