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Question
Solve the following determinant equation:
Solution
Let Δ `=|(3,-2,sin(3theta)),(-7,8,cos(2theta)),(-11,14,2)|=0`
`=>|(1,-2,sin(3theta)),(1,8,cos(2theta)),(3,14,2)|=0` `["Applying" C_1->C_1+C_2]`
`=>|(1,-2,sin(3theta)),(0,10,cos(2theta)-sin(3theta)),(0,20,2-3sin(3theta))|=0` `["Applying" R_2->R_2-R_1 "and" R_3->R_3-3R_1]`
`=>10(2-3sin(3theta))-20(cos(2theta))-sin(3theta)=0`
`=>20-10sin(3theta)-20cos(2theta)=0`
`=>sin(3theta)+2cos(2theta)-2=0`
`=>3sintheta-4sin^3theta+2-4sin^2theta-2=0`
`=>-sintheta(4sin^2theta+4sintheta-3)=0`
`=>sintheta=0 or 4sin^2theta+4sintheta-3=0`
`=>theta=npi or (2sintheta+3)(2sintheta-1)=0`
`=>theta=npi or sintheta=-3/2 or sintheta=1/2`
`=>theta=npi or theta=npi+(-1)^npi/6,ninZ`
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