If the function x=cos^(2)omegat+bsin^(2)omegat+csinomegatcosomegat represent a simple harmonic motion , then which of the foloowing options can be possible ?

If the function x=cos^(2)omegat+bsin^(2)omegat+csinomegatcosomegat rep

1.	If the function x=cos^(2)omegat+bsin^(2)omegat+csinomegatcosomegat represent a simple harmonic motion , then which of the foloowing options can be possible ?
Answer» a=-b,c=2a a=b,c=-2b a=b=0,c `ne`0 a=b,c=0 Solution :`x = a cos^(2) omegat + b sin^(2) omegat + c sin omegat cos omegat` `= a((1 + cos 2omegat)/(2)) + b((1-cos 2omegat)/(2)) + (1)/(2) c sin 2omegat` (a) : For `a = -b and c= 2a` `x = a cos^(2)omegat - a sin^(2)omegat + 2 a sin omegat cos omegat` ` = a(cos^(2)omegat _ sin^(2)omegat) + a SIN2 omegat` ` = a cos 2omegat + a sin 2omegat` ` = sqrt(2) a sin (2omegat + (PI)/(4))` This represents S.H.M. option (a) is true. (b) : `a= b , c --2b` ` x = b cos^(2)omegat + b sin^(2) omegat - 2b sin omegat cos omegat` ` = b-bsin 2omegat` ` = b(1-sin 2omegat)` ` = b(1-sin2omegat)` This represents S.H.M. of amplitude b. But EQUILIBRIUM occurs at b instead of zero. Option (b) is also true. (c ) a = b = c and `c ne 0` For a = b = 0 `x = (c)/(2) sin2omegat` This represents S.H.M. of option (b) is also true. (d) : a = b and c = 0 `therefore x = b cos^(2)omegat + b sin^(2) omegat = b` This does not represent S.H.M.