x is in the second quadrant means that x/2 is in the first quadrant.
Consider the right triangle drawn in the figure. Let tan(x/2)=a.
Then, let the length of the opposite side to x/2 be a, the adjacent side be 1 and the hypotenuse be square root of a squared +1, as shown in the figure.
sin(x/2)=|opp side|/ |hypotenuse| = [tex] \frac{a}{ \sqrt{ a^{2}+1 } } [/tex]
cos (x/2) = |adj side|/ |hypotenuse| = [tex] \frac{1}{ \sqrt{ a^{2}+1 } } [/tex]
from the famous identity: sin(2a)=2sin(a)cos(a), we have:
2sin(x/2)cos (x/2)=sin(x)
thus
[tex]2* \frac{a}{ \sqrt{ a^{2}+1 } }*\frac{1}{ \sqrt{ a^{2}+1 } }= \frac{3}{5} [/tex]
[tex]2* \frac{a}{ a^{2}+1 }= \frac{3}{5} [/tex]
[tex]10a=3 a^{2} +3[/tex]
[tex]3 a^{2}-10a +3=0[/tex]
(3a-1)(a-3)=0
thus a=1/3 or a=3
thus tan(x/2)=1/3 or tan(x/2)=3
Answer: {1/3, 3}
