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Unit 8 Inverse Trigonometric Functions & Graphs Solved Notes Class 12 FBISE New NBF Book 2024-25 PDF | Hamdard Science Academy
Best unit 8 inverse trigonometric functions solved notes class 12 fbise – 100% solved Exercise 8.1 to 8.3 + Full Review Exercise with principal values, graphs & FBISE marking scheme from new nbf book inverse trig unit 8.
Perfect for FBISE HSSC-II April 2026 Exams – score 90+ marks! Ideal for students, teachers, lecturers at FBISE institutions like Angelique School, Fazaia Colleges, Army Public Schools.
Exercise 8.1 – Inverse Trig Evaluations (All Questions Solved) Exercise 8.1 Page 539-542 Full Solutions with Values ↳ Exercise 8.1 Complete Solved + Principal Ranges PDF ↳ Standard Angles & Trig Values 0, π/6, π/4, π/3, π/2 ↳ Principal Ranges sin⁻¹ [-π/2, π/2], cos⁻¹ [0, π] ↳ Q1(i) cos⁻¹(√3/2) = π/6 ↳ Q1(ii) sin⁻¹(1) = π/2 ↳ Q1(iii) tan⁻¹(√3) = π/3 ↳ Q1(iv) cot⁻¹(√3/3) = π/3 ↳ Q1(v) sec⁻¹(2/√3 * 3/3) = π/6 ↳ Q1(vi) csc⁻¹(-√2) = -π/4 ↳ Q1(vii) cos⁻¹(-1/2) = 2π/3 ↳ Q1(viii) tan⁻¹(1/√3) = π/6 ↳ Q1(ix) csc⁻¹(-2) = -π/6 ↳ Sum Values tan⁻¹(1) + cos⁻¹(-1/2) + sin⁻¹(-1/2) = 3π/4
Exercise 8.2 – Graphs of Inverse Trig Functions ↳ Exercise 8.2 Full Solutions + Graphs ↳ y = arccos(x²) Domain [-1,1] Graph ↳ y = arcsin(2x) Domain [-1/2,1/2] Graph ↳ y = arctan(x²) Graph ↳ [y = arccsc(-x) Domain (-\infty,-1] U 1,\infty) Graph ↳ y = arccos(2x-1) Domain [0,1] Graph ↳ y = arctan(3x+1) Graph ↳ y = arccot(x² -1) Graph ↳ y = arcsin(-x) Domain [-1,1] Graph ↳ y = arccos(-2x) Domain Subset [-1,1] Graph ↳ y = arccot(2x-1) Graph
Exercise 8.3 – Identities & Equations (100% Solved) ↳ Exercise 8.3 Identities + Proofs Solved ↳ sin⁻¹(1/3) + sin⁻¹(2√2/3) = π/2 Proof ↳ cos⁻¹(3/5) + cos⁻¹(4/5) = π/2 Proof ↳ tan⁻¹(3/4) + tan⁻¹(1/7) = π/4 Proof ↳ cot⁻¹(2+√3) + cot⁻¹(2-√3) = π/2 Proof ↳ csc⁻¹(√2) + csc⁻¹(√2) = π/2 Proof ↳ sec⁻¹(2) + sec⁻¹(2) = 2π/3 Proof ↳ sin⁻¹(x) + sin⁻¹(-x) = 0 Proof ↳ cos⁻¹(x) + cos⁻¹(-x) = π Proof ↳ cot⁻¹(1) – cot⁻¹(√3) = π/12 Proof ↳ csc⁻¹(√2) – csc⁻¹(2) = π/12 Proof ↳ cos⁻¹(1/2) – cos⁻¹(√3/2) = π/6 Proof ↳ sec⁻¹(x+1/x) + sec⁻¹(x-1/x) = π for x=0 ↳ csc⁻¹(2x+1/x) + csc⁻¹(x+1/x) = 0 for x=0 ↳ cot⁻¹(x-1) + cot⁻¹(2x-1) = -π/4 for x=±1 ↳ sin⁻¹(x-1) = π/2 → x=2 ↳ sin⁻¹x = cos⁻¹(4/5) → x=3/5 ↳ tan⁻¹x = sin⁻¹(24/25) → x=24/7 ↳ cos⁻¹(x-√3/2) = π/6 → x=√3 ↳ tan⁻¹(x +√2/2) = π/4 → x=1 – √2/2 ↳ cos⁻¹(x-1/2) = π/3 → x=1 ↳ 6 ArcCot(2x) -5π = 0 → x=-√3/2 ↳ 9 ArcSin²(x) -π² = 0 → x=±√3/2 ↳ π -2 sin⁻¹x = 2π → x=-1 ↳ 4 ArcTan²(x) -3π ArcTan(x) -π² = 0 → x=-1 ↳ tan^{-1} (√(1+cos x)+√(1-cos x)/√(1+cos x)-√(1-cos x)) = π/4 – x/2, x∈(0,π/2) → No solution
Bonus Resources ↳ Inverse Trig Formulas Cheat Sheet PDF ↳ Top 30 Principal Value Shortcuts ↳ Inverse Trig Graphs Video Series ↳ Unit 8 Past Papers Solved (2018-2025) ↳ Unit 7 Conic Section Notes ↳ Unit 9 Trigonometric Identities Coming Soon
View Full 61-Page Original Solved PDF (No Download Needed) → Open Complete Unit 8 Inverse Trig Solved Notes PDF Now
External Links
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