Find the value of `theta` if `|[1,1,sin 3theta] , [-4,3,cos 2theta] , [7, -7, -2]|=0`

Answer

N/a

Explanation

We have `|(1,1,sin3 theta),(-4,3,cos2theta),(7,-7,-2)|=0`
`implies |(0,1,sin3 theta),(-7,3,cos2theta),(14,-7,-2)|=0 [ :' C_(1)toC_(1)-C_(2)]`
`implies 7|(0,1,sin3theta),(-1,3,cos2theta),(2,-7,-2)|=0` [taking 7 common from `C_(1)`]
`implies 7[0-1(2-2cos2theta)+sin3theta(7-6)]=0` [expanding along `R_(1)`]
`=7[-2(1-cos2theta)+sin3theta]=0`
`-14+14cos2theta+7sin3theta=0`
`implies14cos2theta+7sin3theta=14`
`implies 14(1-2sin^(2)theta)+7(3sin theta-4sin^(3) theta)=14`
`-28sin^(2) theta+14+21sin theta-28 sin^(3) theta=14`
`=-28 sin^(2) theta-28 sin^(3) theta+21 sin theta =0`
`implies28sin^(3) theta+28 sin^(3) theta-21 sin theta=0`
`implies 4 sin^(3) theta+4sin^(2) theta-3 sin theta=0`
`implies sin theta(4 sin^(2)theta+4sin theta-3)=0`
`implies` Either `sin theta=0` ltbgt `implies theta =npi` or `4sin^(2) theta+4sin theta-3=0`
`:. sin theta=(-4+-sqrt(16+48))/8=(-4-sqrt(64))/8`
`=(-4+-8)/8=4/8, (-12)/8`
`sin theta=1/2,(-3),2`
If `sin theta=1/2="sin" (pi)/6`, then
`=theta =npi+(-1)^(n)(pi)/6`
Hence `sin theta=(-3)/2` [not possible becaue `-1le sin theta le 1`]