Find `sin``x/2``,cos``x/2` and `tan``x/2`of the following : `sin x =1/4,`x in quadrant II.

Explanation

We know that,
`90^@lt x lt 180^@`
`45^@ lt x/2 lt 90^@` , So `x/2` in 1st quadrant
As, `sinx=1/4`
`cosx=-sqrt(1-sin^2x)`, because `90^@ lt x lt 180^@`
`cosx=-sqrt(1-(1/4)^2)=-sqrt(15/16)=-sqrt(15)/4`
`cosx=1-2sin^2(x/2)=-sqrt(15)/4`
`2sin^2(x/2)=1+sqrt(15)/4=(4+sqrt(15))/4`
`sin^2(x/2)=(4+sqrt(15))/8`
`sin(x/2)=sqrt((4+sqrt(15))/8)=sqrt((8+2sqrt(15))/(16))`
`sin(x/2)=sqrt((sqrt(5)+sqrt(3))^2)/4=(sqrt(5)+sqrt(3))/4`
`cos(x/2)=+sqrt(1-sin^2(x/2))=sqrt(1-(4+sqrt(15))/8)`

`cos(x/2)=(sqrt(5)-sqrt(3))/4`
`tan(x/2)=sin(x/2)/cos(x/2)=((sqrt(5)+sqrt(3))/4)/((sqrt(5)-sqrt(3))/4)`
After Solving, we get
`tan(x/2)=4+sqrt(15)`