The value of that <span>(√5 + 1)/8 and We can prove it like this: </span>cos 2A= 2cos^2 A-1=1-2sin^2 A <span>so cos^2 48 – sin^2 12 </span> <span>=(1/2)(1+cos 96) - (1/2)(1-cos24) </span> <span>=(1/2)(cos 96 +cos 24) and using cos(A)+cosB)=2cos((A+B)/2)cos((A-B)/2) </span> <span>=cos60cos36 </span> <span>=(1/2)cos36 </span> <span>and you have to find cos36. </span> <span>Suppose 5x=180, then </span> <span>cos(5x)=cos(180) then x=-180,-108, -36, 36, 108 </span> <span>Also 3x=180-2x </span> <span>so cos(3x)=cos(180-2x)=-cos2x </span> <span>so 4cos^3(x) -3cos(x)=1 - 2cos^2(x) </span> <span>giving 4c^3+2c^2 - 3c-1=0 where c=cosx </span> <span>c=-1 is a root and factorizing gives </span> <span>(c+1)(4c^2-2c-1)=0 </span> <span>so 4c^-2c-1=0 giving </span> <span>c=(2±√20)/8=(1±√5)/4 and these have values cos(36) and cos(108) </span> <span>the positive root is therefore cos(36)=(1+√5)/4 </span> <span>and the required value (1/2)cos(36)=(1+√5)/8 </span>I think this can be very useful
Rational numbers include all integers and fractions. All negative integers and whole numbers make up the set of integers. Whole numbers comprise of all natural numbers and zero.
