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Principal Values

It is at times necessary to consider inverse trigonometric functions as single valued. To do this, we select only one value out of many values of angles corresponding to the given value of x. This selected value is called the principal value.
For sin θ = x or θ = sin–1 x, among all values of θ satisfying this relation, there is only one value between –π/2 and π/2. This value of θ is called the principal value of sin–1x. For example, for sin–1 x = 111720.png, the principal value is π/4. If x is +ve, it lies between 0 and π/2 and if x is –ve it lies between –π/2 and 0. Therefore, the principal value of sin–1(–1/2) is –(π/6).

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