How do you find the exact value of cos (theta/2), given that csc theta = - 3/2 and tan theta > 0?

1 Answer
Jan 2, 2017

Answer:

#cos(theta/2)=-(sqrt15-sqrt3)/6#

Explanation:

As #csctheta=-3/2# and #tantheta>0#

we have #sintheta=1/(-3/2)=-2/3#

As sine ratio is negative and tangent ratio is positive, #theta# lies in Q3 and hence #costheta# will be negative and as #pi < theta < (3pi)/2#, we have

#pi/2 < theta/2 < (3pi)/4# and #cos(theta/2)# is too negative.

and #costheta=sqrt(1-(-2/3)^2)=sqrt(1-4/9)=-sqrt5/9=-sqrt5/3#

and as #costheta=2cos^2(theta/2)-1#

#cos(theta/2)=sqrt((1+costheta)/2)==sqrt((1-sqrt5/3)/2)#

= #-sqrt((3-sqrt5)/6)=-sqrt((18-6sqrt5)/36)#

= #-sqrt((15+3-2sqrt45)/36)#

= #-sqrt(((sqrt15)^2+(sqrt3)^2-2sqrt(15xx3))/36)#

= #-(sqrt15-sqrt3)/6#