The value of the determinant `|[sintheta, costheta, sin2theta] , [sin(theta+(2pi)/3), cos(theta+(2pi)/3), sin(2theta+(4pi)/3)] , [sin(theta-(2pi)/3), cos(theta-(2pi)/3), sin(2theta-(4pi)/3)|`

|(sin theta, cos theta, sin2theta),(sin(theta+(2pi)/(3)),cos(theta+(2pi)/(3)),sin(2theta+(4pi)/(3))),(sin(theta-(2pi)/(3)),cos(theta-(2pi)/(3)),sin(2theta-(4pi)/(3)))|=

The value of the determinant |sin theta,cos theta,sin2 thetasin(theta+(2 pi)/(3)),cos(theta+(2 pi)/(3)),sin(2 theta+(4 pi)/(3))sin(theta-(2 pi)/(3)),cos(theta-(2 pi)/(3)),sin(2 theta-(4 pi)/(3))

Prove that all values of theta: |(sintheta, costheta, sin2theta),(sin(theta+(2pi)/3), cos(theta+(2pi)/3), sin (2theta+(4pi)/3)),(sin (theta- (2pi)/3), cos (theta- (2pi)/3), sin (2theta- (4pi)/3))|=0

By using properties of determinats. Prove that- |(sintheta,costheta,sin2theta),(sin(theta+(2pi)/3), cos(theta + (2pi)/3) ,sin(2theta + (4pi)/3)),(sin(theta-(2pi)/3) ,cos(theta - (2pi)/3), sin (2theta - (4pi)/3))|= 0

Prove that for all values of theta , |{:(sintheta,costheta,sin2theta),(sin(theta+(2pi)/(3)),cos(theta+(2pi)/(3)),sin(2theta+(4pi)/(3))),(sin(theta-(2pi)/(3)),cos(theta-(2pi)/(3)),sin(2theta-(4pi)/(3))):}|=0

Prove that: 4sin theta sin(theta+(pi)/(3))sin(theta+(2 pi)/(3))=sin3 theta

If x sintheta=ysin(theta+(2pi)/(3))=z sin(theta+(4pi)/(3)), then

The volume of the parallelepiped whose coterminous edges are represented by the vectors 2vecb xx vecc, 3vecc xx veca and 4veca xx vecb where veca=(1+sintheta)hati+costhetahatj+sin2thetahatk , vecb=sin(theta+(2pi)/(3))hati+cos(theta+(2pi)/(3))hatj+sin(2theta+(4pi)/(3))hatk , vecc=sin(theta-(2pi)/(3))hati+cos(theta-(2pi)/(3))hatj + sin(2theta-(4pi)/(3))hatk is 18 cubic units, then the values of theta , in the interval (0,pi/2) , is/are

The volume of the parallelepiped whose coterminous edges are represented by the vectors 2vecb xx vecc, 3vecc xx veca and 4veca xx vecb where veca=(1+sintheta)hati+costhetahatj+sin2thetahatk , vecb=sin(theta+(2pi)/(3))hati+cos(theta+(2pi)/(3))hatj+sin(2theta+(4pi)/(3))hatk , vecc=sin(theta-(2pi)/(3))hati+cos(theta-(2pi)/(3))hatj + sin(2theta-(4pi)/(3))hatk is 18 cubic units, then the values of theta , in the interval (0,pi/2) , is/are

The volume of the parallelepiped whose coterminous edges are represented by the vectors 2vecb xx vecc, 3vecc xx veca and 4veca xx vecb where veca=(1+sintheta)hati+costhetahatj+sin2thetahatk , vecb=sin(theta+(2pi)/(3))hati+cos(theta+(2pi)/(3))hatj+sin(2theta+(4pi)/(3))hatk , vecc=sin(theta-(2pi)/(3))hati+cos(theta-(2pi)/(3))hatj + sin(2theta-(4pi)/(3))hatk is 18 cubic units, then the values of theta , in the interval (0,pi/2) , is/are