Use the given information to determine the values of sin the

Use the given information to determine the values of sin theta, cos theta, and tan theta.

tan 2theta=7/13; pi<2theta<3pi/2

given

tan 2theta=7/13

since we know tan (2 x ) = 2 tanx / 1 - tan^2 (x)

then 2 tan( theta) / 1 - tan^2 (theta) = 7/13

==> 2 tan( theta) = ( 1 - tan^2 (theta) ) 7

==> 13 ( 2 tan( theta) ) = 7 - 7 tan^2 (theta)

==> 26 tan( theta ) = 7 - 7 tan^2 (theta)

==> 7 tan^2 (theta) - 26 tan( theta ) - 7 =0

we get

tan ( theta ) = 13 + sqrt( 218 ) / 7 or 13 - sqrt( 218 ) / 7

tan ( theta ) = opposite side / adjacent side

==> 13 + sqrt( 218 ) / 7

opposite side = 13 + sqrt( 218 ) ==> 27.76482

adjacent side = 7

hypotnuese = sqrt ( opposite side^2 + adcent side^2 )

==> sqrt ( (13 + sqrt( 218 ))^2 + 7^2 )==> 28.63364

sin theta = opp / hyp ==> 27.76482 / 28.63364

cos theta = adj / hyp ==> 7 / 28.63364