Learn Exercise 2.2 with Free Lessons & Tips

We know that cos⁻¹ (cos x) = x if, which is the principal value branch of cos ⁻¹x.

Here,

Now, can be written as:

cos−1(cos7π6) = cos−1[cos(π+π6)]cos−1(cos7π6) = cos−1[− cosπ6]             [as, cos(π+θ) = − cos θ]cos−1(cos7π6)  = cos−1[− cos(π−5π6)]cos−1(cos7π6) = cos−1[−{− cos (5π6)}]   [as, cos(π−θ) = − cos θ]cos-1cos7π6 = cos-1cosπ+π6cos-1cos7π6 = cos-1- cosπ6             as, cosπ+θ = - cos θcos-1cos7π6  = cos-1- cosπ-5π6cos-1cos7π6 = cos-1-- cos 5π6   as, cosπ-θ = - cos θ

The correct answer is B.

1.

Ans. 

=

2.

Now, can be written as:

3. 

Ans.

Let. Then,

We know that the range of the principal value branch of

Let.

The range of the principal value branch of

The correct answer is B.

Answer is  D)1 

Since sin inverse of 0.5 is -30 degree.Since sin(60-(-30)) = sin 90 =1

(1)To prove:

Let x = sinθ. Then,

We have,

R.H.S. =

= 3θ

= L.H.S.

(2)To prove:

Let x = cosθ. Then, cos⁻¹x =θ.

We have,

(3)To prove:

(4)To prove:

1.

2.

Put x = cosec θ ⇒ θ = cosec⁻¹x

3.

4.

5.

6.