Why restrict the domain? sin x is not one-oneप्रांत सीमित क्यों? sin x एकैकी नहीं हैDomain restrict kyon? sin x one-one nahi hai

NCERT Section 2.2 · Class 12 Maths · Chapter 2 · interactive diagram

Parts and what they mean

• y = \sin x on all of \mathbb{R} (पूरे \mathbb{R} पर y = \sin x) — Domain: all real numbers, range: [-1, 1]. The wave repeats every 2\pi.
• Horizontal line y = \tfrac12 (क्षैतिज रेखा y = \tfrac12) — It cuts the full sine graph at many points: many x have the same sine, so sin is not one-one and has no inverse on \mathbb{R}.
• x = \pi/6 (x = \pi/6) — Inside the band [-\pi/2, \pi/2] exactly one x has \sin x = \tfrac12. So \sin^{-1}\tfrac12 = \pi/6.
• Other solutions 5\pi/6,\ -7\pi/6, \dots (अन्य हल 5\pi/6,\ -7\pi/6, \dots) — These also give \sin x = \tfrac12 but lie outside the band, so they are not the principal value.
• Restricted domain [-\pi/2, \pi/2] (सीमित प्रांत [-\pi/2, \pi/2]) — On this band sin is one-one and onto [-1, 1], so it can be reversed.
• x-axis (x-अक्ष) — The horizontal number line; inputs x are read here.
• y-axis (y-अक्ष) — The vertical number line; outputs y = f(x) are read here.
• Origin O(0, 0) (मूल बिंदु O(0, 0)) — Where the two axes cross.

Original diagram drawn by Simple Learning Hub, based on NCERT Section 2.2. Report a mistake