If tanx =3/4, π < x <3 π /2, find the values of sin (x/2), cos (x/2) and tan (x/2)?

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Answer 1 · 2016-09-28T16:49:36

Given #pi < x<(3pi)/2 and tanx=3/4#

#pi < x<(3pi)/2#

#=>pi/2 < x/2<(3pi)/4->x/2in " 2nd quadrant"#

This means

#sin(x/2)->+ve#

#cos(x/2)->-ve#

#tan(x/2)->-ve#

Now #tanx=3/4#

#=>(2tan(x/2))/(1-tan^2(x/2))=3/4#

#=>8tan(x/2)=3-3tan^2(x/2)#

#=>3tan^2(x/2)+8tan(x/2)-3=0#

#=>3tan^2(x/2)+9tan(x/2)-tan(x/2)-3=0#

#=>3tan(x/2)(tan(x/2)+3)-1(tan(x/2)+3)=0#

#=>(3tan(x/2)-1)(tan(x/2)+3)=0#

This means

#tan(x/2)=1/3->"not acceptable as "tan(x/2)->-ve#

So #tan(x/2)->-3#

Now

#cos(x/2)=1/sec(x/2)=-1/sqrt(1+tan^2(x/2)#

#=-1/sqrt(1+(-3)^2)=-1/sqrt10#

Again

#sin(x/2)=tan(x/2)xxcos(x/2)#

#=-3xx(-1/sqrt10)=3/sqrt10#