#### Topics

1. Find the principal value of sin-1 (-1/2)?

Ans : Let sin-1 (-1/2) = y

Then sin y = -1/2

= - sin (π/6) = sin (/6 )

We know that the range of the principal value branch of sin-1 is

[/2 , /2 ] and sin [/6 ] = -1/2

∴  the principal value of sin-1 (-1/2) is - π/6.

2. Find the principal value of cos-1 (√3/2)?

Ans : Let cos-1 (√3/2) = y.

Then , cos y = √3/2 = cos (π/6 ).

We know that the range of the principal value branch of cos-1 is [ 0 , π ] and cos (π/6 ) = √3/2.

Therefore, the principal value of cos-1 (√3/2) is π/6.

3. Find the principal value of tan-1 (-1)?

Ans : Let tan-1 (-1) = y

Then , tan y = -1 = - tan (π/4) = tan (- π/4 )

We know that the range of the principal value branch of tan-1 is

[/2 , π/2 ] and tan (/4) = -1

Therefore, the principal value of tan-1(-1) is /4

4. Find the value of cos-1(1/2) + 2 sin-1 (1/2) ?

Ans : Let cos-1(1/2) = x

Then , cos x = 1/2 = cos (π/3)

Cos-1 (1/2) = π/3

Let sin-1 (1/2) = y. Then sin y = 1/2 = sin (π/6).

Sin-1 (1/2) = π/6

cos-1(1/2) + 2 sin-1 (1/2) = π/3 + /6 = π/3 + π/3 = /3

5. Prove 3 sin-1 x = sin-1 (3x - 4x3), x ∈ [-1/2, 1/2 ]?

Ans : To prove 3 sin-1 x = sin-1 (3x - 4x3), x ∈ [-1/2, 1/2 ]

Let x = sin 6

Then , sin-1 x = 6

We have,

R.H.S = sin-1 (3x - 4x3 )

= sin-1 (3 sin6 - 4 sin36 )

= sin-1 (sin 36)

= 36

= 3 sin-1 (x)

= L .H .S

6. Prove tan-1 2/11 + tan-1 7/24 = tan-1 1/2?

Ans : L.H .S = tan-1 2/11 + tan-1 7/24 7. Write the function in the simplest form : Ans : 8. Write the function in the simplest form : Ans : 9. Find the value of cos-1 (cos 13π/0 ) ?

Ans : We know that cos-1(cos x ) = x if x∈ [ 0 , π ] , which is the principal value branch of cos-1x.

Here, 13/6 ∉  [ 0 , π ]

Now, cos-1 (cos 13π/6 ) can be written as

cos-1 (cos 13π/6 ) = cos-1 [ cos (2 π + π/6 ) ]

= cos-1 [ cos ( π/6) ], where π/6 ∈ [ 0 , π ]

∴    cos-1 (cos 13π/6 ) = cos-1 [ cos ( π/6) ] = π/6

10. Prove that 2 sin-1 3/5 = tan-1 24/7?

Ans : Let sin-1 3/5 = x

Then sinx = 3/5 ∴  tanx = 3/4

∴  x = tan-1 3/4

⇒  sin-1 3/5 = tan-1 3/4

Now , we have:

L.H.S = 2 sin-1 3/5 = 2 tan-1 3/4 Paid Users Only!
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