What is the maximum value of $p$ ?

Question

Question

Consider the following for the two (02) items that follow: Let

p = ∣ sin α - sin (α - 9 0^{\circ}) ∣

.

This question was previously asked in

NDA-I (Mathematics) Official Paper (Held On: 13 Apr, 2025)

Answer 1

Calculation:

Given,

The expression for p is:

$ p = |\sin \alpha - \sin(\alpha - 90^\circ)| $

Using the identity for $\sin(\alpha - 90^\circ) $

$ \sin(\alpha - 90^\circ) = \cos \alpha $

Substituting this into the expression for p :

$ p = |\sin \alpha - \cos \alpha| $

The expression $|\sin \alpha - \cos \alpha|$ achieves its maximum value when the difference between $\sin \alpha $ and $\cos \alpha $ is largest.

We can rewrite $\sin \alpha - \cos \alpha $ as:

$ \sin \alpha - \cos \alpha = \sqrt{2} \left( \sin \left( \alpha - 45^\circ \right) \right) $

Since $\sin(\alpha - 45^\circ) $ has a maximum value of 1, the maximum value of $\sqrt{2} \left( \sin(\alpha - 45^\circ) \right) $ is:

$ \sqrt{2} $

Hence, the correct answer is Option 2

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