Unit circle: why sin⁻¹(½) = π/6 and not 5π/6इकाई वृत्त: sin⁻¹(½) = π/6 क्यों, 5π/6 क्यों नहींUnit circle: sin⁻¹(½) = π/6 kyon, 5π/6 kyon nahi

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

Parts and what they mean

• Unit circle (radius 1) (इकाई वृत्त (त्रिज्या 1)) — A point at angle θ on it is (\cos\theta, \sin\theta): x-coordinate = cos, y-coordinate = sin.
• x-axis (cos values) (x-अक्ष (cos के मान)) — The x-coordinate of a point on the circle is \cos\theta.
• y-axis (sin values) (y-अक्ष (sin के मान)) — The y-coordinate of a point on the circle is \sin\theta.
• Centre O (केंद्र O) — Angles are measured from the positive x-axis, anticlockwise = positive.
• The \sin^{-1} zone: [-\pi/2, \pi/2] (\sin^{-1} क्षेत्र: [-\pi/2, \pi/2]) — The right half of the circle (from −90° to 90°). Every value of \sin^{-1} is an angle here.
• Line y = \tfrac12 (रेखा y = \tfrac12) — All angles with \sin\theta = \frac12 sit where this line meets the circle.
• \theta = \pi/6 (30°) — inside the zone (\theta = \pi/6 (30°) — क्षेत्र के अंदर) — So the principal value \sin^{-1}\frac12 = \frac{\pi}{6}.
• \theta = 5\pi/6 (150°) — outside (\theta = 5\pi/6 (150°) — बाहर) — Also \sin\frac{5\pi}{6} = \frac12, but \frac{5\pi}{6} > \frac{\pi}{2}, so it is not the principal value.
• Angle 30° = π/6 (कोण 30° = π/6) — Measured anticlockwise from the positive x-axis.

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