Question

Given cos x = (− 1 2 ) , Find sin x 2 , cos x 2 and tan x 2 having given that x lies in the third quadrant and 0 ≤ x ≤ 2

pls explain as well ​

Answer 1

Answer:

cosx=−

3

1

,π<x<

2

3π

i.e. x lies in 3rd quadrant

Using 1−cosx=2sin

2

2

x

⇒sin

2

x

=±

2

1−cosx

We get, sin

2

x

=±

2

1−(−

3

1

)

=±

6

4

As π<x<

2

3π

⇒

2

π

<

2

x

<

4

3π

and sin is positive in 2nd quadrant

∴sin

2

x

=

5

2

Using 1+cosx=2cos

2

2

x

⇒cos

2

x

=±

2

1+cosx

we get, cos

2

x

=±

2

1+(−

3

1

)

=±

3

1

As π<x<

2

3π

⇒

2

π

<

2

x

<

4

3π

and cos is negative in 2nd quadrant

∴cos

2

x

=−

3

1

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+(−

3

1

)

1−(−

3

1

)

=±

2

As π<x<

2

3π

⇒

2

π

<

2

x

<

4

3π

and tan is negative in 2nd quadrant

∴tan

2

x

=−

2