if given the tan2x, find sin2x without using a double formula

Question

we are given tanx is 3/4. you can use the double formula to find tan2x. the teacher wants us to find 3 other doubles like sin2x or cos2x without using the double angle formulas. how would it be done

Answer 1

Given Tanx = 3/4

Tan2x = 2Tanx/1-Tan^2x

= 2(3/4)/1-(9/16)

= (3/2)/7/16

= 24/7

Sintheta = opposite side/hypotenuse

Costheta = adjacent side/hypotenuse

Tantheta = opposite side/adjacentside = 3/4

opposte side = 3 and adjacent side = 4

Hypotenuse^2 = 3^2+4^2

= 9+16

= 25

Hypotenuse = 5

Sinx = 3/5

Cosx = 4/5

Sin2x = 2SinxCosx = 2(3/5)(4/5)

= 24/25

Cos2x = Cos^2x-Sin^2x

= 16/25-9/25

= 7/25