Trig functions like sin, cos, tan are not one-one & onto
over their full domain — so their inverses don't exist normally!
👉 Trick: restrict the domain → function becomes
bijective (one-one + onto) → now inverse exists 🎉
Each restricted interval → gives a "branch" of the inverse function. The branch we normally use is called the Principal Value Branch 🌿
| Function | Domain | Range (Principal Branch) |
|---|---|---|
| sin⁻¹x | [−1, 1] | [−π/2, π/2] |
| cos⁻¹x | [−1, 1] | [0, π] |
| cosec⁻¹x | R − (−1, 1) | [−π/2, π/2] − {0} |
| sec⁻¹x | R − (−1, 1) | [0, π] − {π/2} |
| tan⁻¹x | R | (−π/2, π/2) |
| cot⁻¹x | R | (0, π) |
✍️ Similar identities hold for cos, tan, cot, sec, cosec — but only within their principal value domains!
Let sin⁻¹(1/√2) = y → sin y = 1/√2
Since range of sin⁻¹ is [−π/2, π/2] and sin(π/4) = 1/√2
👉 Principal value = π/4
cot y = −1/√3 = −cot(π/3) = cot(π − π/3) = cot(2π/3)
Range of cot⁻¹ is (0, π) → 👉 Principal value = 2π/3
3π/5 is NOT in [−π/2, π/2] → can't directly cancel!
But sin(3π/5) = sin(π − 3π/5) = sin(2π/5), and 2π/5 ∈ [−π/2, π/2]
👉 sin⁻¹(sin 3π/5) = 2π/5 (Always adjust to principal range first!)
Whenever the angle given is outside the principal range, DO NOT just cancel sin⁻¹(sin x) = x directly. First bring the angle into the correct range using symmetry formulas (like sin(π − x) = sin x, cos(2π − x) = cos x, etc.) 🧠
Trigonometry began in India! Aryabhata, Brahmagupta, Bhaskara I & II gave major results. The notation sin⁻¹x, cos⁻¹x was suggested by astronomer Sir John F.W. Herschel (1813) ⭐