Question

30
(i) the value of a (ii) sin 30
2
(b) If 2cos? x + sin x - 2 = 0 and 0° <x< 90°, find
(i) the value of x (ii) tan 2x
(iii) cosecx
(iii) 2 cos
.​

Answer 1

Step-by-step explanation:

tanx=−

3

4

,

2

π

<x<π

i.e x lies in 2nd quadrant

Hence tanx=−

3

4

⇒sinx=

4

2

+3

2

4

=

5

4

And cosx=−

4

2

+3

2

5

=−

5

3

Now using 1−cosx=2sin

2

2

x

⇒sin

2

x

=±

2

1−cosx

, we get

sin

2

x

=±

2

1−(−

5

3

)

=±

10

8

As

2

π

<x<π⇒

4

π

<

2

x

<

2

π

and sine is positive in 1st quadrant

Then 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+(−

5

3

)

=±

10

2

2

π

<x<π⇒

4

π

<

2

x

<

2

π

and cos is positive 1st quadrant

∴cos

2

x

=

5

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+(−

5

3

)

1−(−

5

3

)

=±

4

As

2

π

<x<π⇒

4

π

<

2

x

<

2

π

and tan is positive in 1st quadrant

∴tan

2

x

=2

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