Question

find the principal value of cos‐¹(-1)​

Answer 1

Answer:

We will learn how to find the principal values of inverse trigonometric functions in different types of problems.

The principal value of sin\(^{-1}\) 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 sine is x. For this reason sin^-1 x is also denoted by arc sin x. Similarly, cos\(^{-1}\) x, tan\(^{-1}\) x, csc\(^{-1}\) x, sec\(^{-1}\) x and cot\(^{-1}\) x are denoted by arc cos x, arc tan x, arc csc x, arc sec x.

1. Find the principal values of sin\(^{-1}\) (- 1/2)

Solution:

If θ be the principal value of sin\(^{-1}\) x then - \(\frac{π}{2}\) ≤ θ ≤ \(\frac{π}{2}\).

Therefore, If the principal value of sin\(^{-1}\) (- 1/2) be θ then sin\(^{-1}\) (- 1/2) = θ

⇒ sin θ = - 1/2 = sin (-\(\frac{π}{6}\)) [Since, - \(\frac{π}{2}\) ≤ θ ≤ \(\frac{π}{2}\)]

Therefore, the principal value of sin\(^{-1}\) (- 1/2) is (-\(\frac{π}{6}\)).