find sin x/2, cos x/2 and tan x/2 in exact form if sin x=3/5 and 0<x<pi/2

Question

I need to use appropriate  trigonometric formulae and identities to answer this question

Answer 1

sin(x)=⅗, so we have a 3-4-5 Pythagorean Triangle where 5 is the length of the hypotenuse.

cos(x)=⅘. (Pythagoras: 5²=3²+4².) tan(x)=¾ or tan(x)=sin(x)/cos(x)=⅗/⅘=¾.

Let t=t/1=tan(x/2), tan(x)=2t/(1-t²), sin(x/2)=t/√(1+t²), cos(x/2)=1/√(1+t²) from right triangle with sides 1, t, √(1+t²) (Pythagoras).

tan(x)=2t/(1-t²)⇒¾(1-t²)=2t, 3(1-t²)=8t, 3t²+8t-3=0=(3t-1)(t+3), so t=tan(x/2)=1/3 or -3. We can reject -3,  because x would not be in the required range 0-π/2. Therefore, t=tan(x/2)=⅓, sin(x/2)=⅓/√(10/9)=1/√10 which is the same as √10/10. cos(x/2)=1/√(10/9)=3/√10 or 3√10/10 or 0.3√10.