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Must-know facts

16 facts

1No trigonometric function is one-one on its natural domain; each is cut to a principal interval before inverting.
2sin⁻¹ : [−1, 1] → [−π/2, π/2]; cos⁻¹ : [−1, 1] → [0, π].
3tan⁻¹ : R → (−π/2, π/2); cot⁻¹ : R → (0, π).
4cosec⁻¹ : R − (−1, 1) → [−π/2, π/2] − {0}; sec⁻¹ : R − (−1, 1) → [0, π] − {π/2}.
5sin⁻¹ x is an angle; (sin x)⁻¹ = 1/sin x. They are different things.
6Principal value = the one value inside the principal range.
7Negative input: sin⁻¹, tan⁻¹, cosec⁻¹ give a negative angle; cos⁻¹, cot⁻¹, sec⁻¹ give an angle in (π/2, π].
8sin⁻¹(1/√2) = π/4; cot⁻¹(−1/√3) = 2π/3; cos⁻¹(−1/2) = 2π/3; tan⁻¹(−1) = −π/4.
9The 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

Giving cos⁻¹(−1/2) as −π/3.

cos⁻¹ returns angles in [0, π]. cos⁻¹(−1/2) = π − π/3 = 2π/3.

Writing sin⁻¹(sin 2π/3) = 2π/3.

2π/3 is outside [−π/2, π/2]. Swap to the principal angle with the same sine: π/3.

Reading sin⁻¹ x as 1/sin x.

sin⁻¹ x is the angle whose sine is x. 1/sin x is cosec x, written (sin x)⁻¹.

Evaluating sin⁻¹ 2 or sec⁻¹(1/2).

sin⁻¹ and cos⁻¹ need |x| ≤ 1; sec⁻¹ and cosec⁻¹ need |x| ≥ 1. Outside those, the expression is undefined.

Using cot⁻¹ x = tan⁻¹(1/x) for negative x.

It holds only for x > 0. cot⁻¹(−1) = 3π/4, but tan⁻¹(−1) = −π/4.

Applying 2 sin⁻¹ x to sin⁻¹(2x√(1 − x²)) for every x.

That form holds only for |x| ≤ 1/√2. For 1/√2 ≤ x ≤ 1 the answer is 2 cos⁻¹ x.

Using tan⁻¹ x + tan⁻¹ y = tan⁻¹((x + y)/(1 − xy)) when xy > 1.

With x, y > 0 and xy > 1 the true sum exceeds π/2, so add π to the right side: tan⁻¹ 2 + tan⁻¹ 3 = π + tan⁻¹(−1) = 3π/4.

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.