📚 Key Concepts
Principal Value Branch- sin⁻¹x: Domain [-1,1], Range [-π/2, π/2]
- cos⁻¹x: Domain [-1,1], Range [0, π]
- tan⁻¹x: Domain R, Range (-π/2, π/2)
- cot⁻¹x: Domain R, Range (0, π)
- sec⁻¹x: Domain (-∞,-1]∪[1,∞), Range [0,π]-{π/2}
- cosec⁻¹x: Domain (-∞,-1]∪[1,∞), Range [-π/2,π/2]-{0}
Relationships- sin⁻¹(1/x) = cosec⁻¹x
- cos⁻¹(1/x) = sec⁻¹x
- tan⁻¹(1/x) = cot⁻¹x
🎯 Key Formulas
For |x| ≤ 1:- sin(sin⁻¹x) = x
- cos(cos⁻¹x) = x
For all x in domain:- sin⁻¹(sin x) = x for x ∈ [-π/2, π/2]
- cos⁻¹(cos x) = x for x ∈ [0, π]
Important identities:- sin⁻¹x + cos⁻¹x = π/2
- tan⁻¹x + cot⁻¹x = π/2
- sin⁻¹x + sin⁻¹y = sin⁻¹[x√(1-y²) + y√(1-x²)]
⚠️ Common Mistakes to Avoid
Forgetting domain restrictionsMixing up ranges of different functionsIncorrect sign in composite functionsNot considering quadrant for solutionsWrong application of inverse function properties
📖 Knowledge Prerequisites
- Basic trigonometry
- Understanding of functions
- Knowledge of coordinate geometry
- Concept of inverse functions
💡 Tips for Students
Always check domain and rangeDraw graphs to visualize functionsRemember principal value rangesPractice with both degrees and radiansUse unit circle for reference
👉 Practice Recommedations
Find values of inverse trig expressionsSolve equations involving inverse functionsVerify identitiesGraph inverse trig functionsPractice composition of functionsSolve problems involving applications
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