Lesson 12 of 12 · 13 min
Chapter review
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JEE-level walkthrough, definitions to simplification
Must-know facts
16 facts
- 01No trigonometric function is one-one on its natural domain; each is cut to a principal interval before inverting.
- 02sin⁻¹ : [−1, 1] → [−π/2, π/2]; cos⁻¹ : [−1, 1] → [0, π].
- 03tan⁻¹ : R → (−π/2, π/2); cot⁻¹ : R → (0, π).
- 04cosec⁻¹ : R − (−1, 1) → [−π/2, π/2] − {0}; sec⁻¹ : R − (−1, 1) → [0, π] − {π/2}.
- 05sin⁻¹ x is an angle; (sin x)⁻¹ = 1/sin x. They are different things.
- 06Principal value = the one value inside the principal range.
- 07Negative input: sin⁻¹, tan⁻¹, cosec⁻¹ give a negative angle; cos⁻¹, cot⁻¹, sec⁻¹ give an angle in (π/2, π].
- 08sin⁻¹(1/√2) = π/4; cot⁻¹(−1/√3) = 2π/3; cos⁻¹(−1/2) = 2π/3; tan⁻¹(−1) = −π/4.
- 09The graph of an inverse is the graph of the trimmed function reflected in y = x.
- 10sin(sin⁻¹ x) = x on [−1, 1]; sin⁻¹(sin x) = x only on [−π/2, π/2].
- 11sin⁻¹(sin 3π/5) = 2π/5; tan⁻¹(tan 3π/4) = −π/4.
- 12sin⁻¹(2x√(1 − x²)) = 2 sin⁻¹ x on [−1/√2, 1/√2] and 2 cos⁻¹ x on [1/√2, 1].
- 13tan⁻¹(cos x/(1 − sin x)) = π/4 + x/2 for −3π/2 < x < π/2.
- 14cot⁻¹(1/√(x² − 1)) = sec⁻¹ x for x > 1.
- 15sin⁻¹ x + cos⁻¹ x = π/2; tan⁻¹ x + cot⁻¹ x = π/2; sec⁻¹ x + cosec⁻¹ x = π/2.
- 16tan⁻¹ x + tan⁻¹ y = tan⁻¹((x + y)/(1 − xy)) for xy < 1.
Common traps
Where marks are lost
Formulas
10 to know
sin⁻¹
sin⁻¹ : [−1, 1] → [−π/2, π/2]
Increasing, odd.
cos⁻¹
cos⁻¹ : [−1, 1] → [0, π]
Decreasing; cos⁻¹(−x) = π − cos⁻¹ x.
tan⁻¹
tan⁻¹ : R → (−π/2, π/2)
Increasing, odd, asymptotes y = ±π/2.
cot⁻¹
cot⁻¹ : R → (0, π)
Decreasing; cot⁻¹(−x) = π − cot⁻¹ x.
cosec⁻¹
cosec⁻¹ : R − (−1, 1) → [−π/2, π/2] − {0}
cosec⁻¹ x = sin⁻¹(1/x).
sec⁻¹
sec⁻¹ : R − (−1, 1) → [0, π] − {π/2}
sec⁻¹ x = cos⁻¹(1/x).
Undoing rules
sin(sin⁻¹ x) = x, x ∈ [−1, 1]; sin⁻¹(sin x) = x, x ∈ [−π/2, π/2]
Likewise for the other five.
Complementary sums
sin⁻¹ x + cos⁻¹ x = tan⁻¹ x + cot⁻¹ x = π/2
Also sec⁻¹ x + cosec⁻¹ x = π/2 for |x| ≥ 1.
tan⁻¹ addition
tan⁻¹ x + tan⁻¹ y = tan⁻¹((x + y)/(1 − xy))
Valid when xy < 1.
Double angle
sin⁻¹(2x√(1 − x²)) = 2 sin⁻¹ x
For −1/√2 ≤ x ≤ 1/√2.
Key terms
4 terms
- Branch
- One choice of restricted interval on which a trigonometric function is inverted.
- Principal value branch
- The agreed branch used whenever no branch is named.
- Principal value
- The value of an inverse trigonometric function that lies in its principal range.
- arc sin
- Another name for sin⁻¹; likewise arc cos, arc tan and so on.