sin⁻¹(sin 3π/5) = 2π/5, seen on the unit circlesin⁻¹(sin 3π/5) = 2π/5, इकाई वृत्त परsin⁻¹(sin 3π/5) = 2π/5, unit circle par

NCERT Example 6 · 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 answer of \sin^{-1} must be on this right half.
• Same height = same sine (समान ऊँचाई = समान sine) — \sin\frac{3\pi}{5} = \sin(\pi - \frac{3\pi}{5}) = \sin\frac{2\pi}{5}.
• 2\pi/5 — the answer (2\pi/5 — उत्तर) — It has the same sine and lies in [-\frac{\pi}{2}, \frac{\pi}{2}], so \sin^{-1}(\sin\frac{3\pi}{5}) = \frac{2\pi}{5}.
• 3\pi/5 — the trap (3\pi/5 — जाल) — \frac{3\pi}{5} > \frac{\pi}{2}, so writing \sin^{-1}(\sin\frac{3\pi}{5}) = \frac{3\pi}{5} is wrong.

Original diagram drawn by Simple Learning Hub, based on NCERT Example 6. Report a mistake