What is the principle value of ?

Let us understand what principal value of inverse trigonometric function is.

The principal value of an inverse trigonometric function, say, cos^-1 x for x > 0, is the length of the arc of a unit circle centred at the origin which subtends an angle at the centre whose cosine is x. For this reason cos^-1 x is also denoted by arc cos x.

Let the principal value of cos^-1 x be θ, such that 0 ≤ θ ≤ π.

Put .

If principal value of be θ, then

We know that, .

But is negative, and cos function is negative in 2^nd and 3^rd quadrant.

Therefore,

…(i)

Or,

…(ii)

Also, we know that 0 ≤ θ ≤ π. So, we need to take case (i).

From case (i),

Hence, principal value of is .