Find the exact value of the given expression sin2 cos1 1517

Find the exact value of the given expression. sin(2 cos^-1 15/17) Evaluate the expression under the given conditions. cos 2 theta; sin theta = -7/25, theta in Quadrant III Prove the identity. sin 8x = 2 sin 4x cos 4x

Solution

13) we have given sin(2*arccos(15/17))

sin(2*arccos(15/17)) =sqrt(sin²(2*arccos(15/17)))

=sqrt(1-cos²(2*arccos(15/17)))

=sqrt(1-(cos(2*arccos(15/17)))²) since cos²(2*arccos(15/17)) =(cos(2*arccos(15/17)))²

⁼sqrt(1-((2cos²(arccos(15/17))-1)²) since cos(2*theta)=2cos²(theta)-1,where theta=arccos(15/17)

=sqrt(1-4cos⁴(arccos(15/17))-1+4cos²(arccos(15/17)))

=sqrt(1-4*(cos(arccos(15/17)))⁴-1+4(cos(arccos(15/17)))²)

=sqrt(-4*(15/17)⁴+4(15/17)²)

=2*(15/17)*sqrt(1-(15/17)²)

=2*(15/17)*sqrt(64/289)

=30/17*8/17

=240/289

sin(2*arccos(15/17)) =240/289

14) we have given cos(2*theta),sin(theta)=-7/25,theta in IIIrd quadrant

cos(2*theta)=1-2sin²(theta) =1-2(-7/25)²=1-98/625=(625-98)/625 =527/625

cos(2*theta)=527/625

15) sin8x=sin(4x+4x)=sin4x*cos4x+cos4xsin4x =2sin4xcos4x since sin(A+B)=sinAcosB+cosAsinB

sin8x=2sin4x cos4x