sin x =

1

4

, x in quadrant II
⇒

π

2

< x < π ⇒

π

4

<

x

2

<

π

2

⇒

x

2

lies in first quadrant
⇒ sin

x

2

> 0 , cos

x

2

> 0 , tan

x

2

> 0
Also , cos2 x = 1 - sin2x ⇒ cos2x = 1 - (

1

4

)2 = 1 -

1

16

=

15

16

⇒ cos x = ±

√15

4

⇒ cos x = -

√15

4

(Since x lies in II quadrant)
cos

x

2

= ± √

1+cosx

2

= ± √

1− √15 4

2

= ± √

4−√15

8

= ±

√8−2√15

4

=

√8−2√15

4

sin

x

2

= ± √

1−cosx

2

= ± √

1+ √15 4

2

= ± √

4+√15

8

=

√8+2√15

4

tan

x

2

=

√8+2√15

√8−2√15

=

√4+√15

√4−√15

×

√4−√15

√4−√15

=

√16−15

4−√15

=

1

4−√15

×

√4+√15

√4+√15

=

√4+√15

1

= 4 + √15
Hence , sin

x

2

=

√8+2√15

4

, cos

x

2

=

√8−2√15

4

, tan

x

2

= 4 + √15