Question

if sinx=-1/2 and x lies in quadrant 4,find the values of 1.sin x/2,2.cos x/2,tan x/2.

Answer 1

Answer:

sinx=

4

1

,x in 2nd quadrant

⇒

2

π

<x<π and cos is negative in 2nd quadrant

Using 1−cosx=2sin

2

2

x

⇒sin

2

x

=±

2

1−cosx

We get, sin

2

x

=±

2

1−(−

4

15

)

=±

8

4+

15

As

2

π

<x<π⇒

4

π

<

2

x

<

2

π

and sin is positive in 1st quadrant

∴sin

2

x

=

4

8+2

15

Using 1+cosx=2cos

2

2

x

⇒cos

2

x

=±

2

1+cosx

We get cos

2

x

=±

2

1+(−

4

15

)

=±

8

4−

15

=±

4

8−2

15

As

2

π

<x<π⇒

4

π

<

2

x

<

2

π

and cos is positive in first quadrant

∴cos

2

x

=

4

8−2

15

Using cosx=

1+tan

2

2

x

1−tan

2

2

x

⇒tan

2

x

=±

1+cosx

1−cosx

We get tan

2

x

=±

1+(−

4

15

)

1−(−

4

15

)

=±

4−

15

4+

15

As

2

π

<x<π⇒

4

π

<

2

x

<

2

π

and tan is positive in first quadrant

∴tan

2

x

=4+

15