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Relations and Functions · Part 14 of 20

Inverse Trigonometric Functions

Trig functions repeat, so each is restricted to a principal branch before inverting. Principal domains and ranges, graphs, negative-argument and reciprocal identities, the π/2 identities, and why sin⁻¹(sin x) is not always x.

Builds on: Part 13 · Domain and Range of Trigonometric Expressions, Part 8 · Inverse Functions.

Inverse Trigonometric Functions

Video coming soon

Principal branch

Invert one piece

On [−π/2, π/2], sin is a bijection onto [−1, 1]. Reflecting that piece in y = x gives $sin^{- 1} : [- 1, 1] \to [- \frac{π}{2}, \frac{π}{2}]$ .

sin x piece and its mirror image. — Grey: the rest of sin x, not used.

Table of principal values. — Learn the six ranges.

Principal values

Domains and ranges

sin⁻¹, tan⁻¹ and cosec⁻¹ have ranges around 0 (odd functions); cos⁻¹, cot⁻¹ and sec⁻¹ have ranges inside [0, π] (f(−x) = π − f(x)).

Identities

Pairs add to π/2

$sin^{- 1} x + cos^{- 1} x = \frac{π}{2}$ , $tan^{- 1} x + cot^{- 1} x = \frac{π}{2}$ , $sec^{- 1} x + cosec^{- 1} x = \frac{π}{2}$ . For x < 0, $tan^{- 1} \frac{1}{x} = - π + cot^{- 1} x$ .

Identities. — Watch the sign of x.

Zig-zag graph. — Shaded: where it equals x.

Composites

sin⁻¹(sin x)

Defined for all real x, but always in [−π/2, π/2]: it equals x only on that interval, and π − x on [π/2, 3π/2].

Summary

Key formulas

sin^{- 1} (- x) = - sin^{- 1} x

Principal domains, ranges and graphs

cos^{- 1} (- x) = π - cos^{- 1} x

Principal domains, ranges and graphs

tan^{- 1} (- x) = - tan^{- 1} x

Principal domains, ranges and graphs

cot^{- 1} (- x) = π - cot^{- 1} x

Principal domains, ranges and graphs

sec^{- 1} (- x) = π - sec^{- 1} x

Principal domains, ranges and graphs

cosec^{- 1} (- x) = - cosec^{- 1} x

Principal domains, ranges and graphs

sin^{- 1} x + cos^{- 1} x = \frac{π}{2}, ∣ x ∣ \leq 1

Useful identities

sec^{- 1} x + cosec^{- 1} x = \frac{π}{2}, ∣ x ∣ \geq 1

Useful identities

tan^{- 1} x + cot^{- 1} x = \frac{π}{2}, x \in R

Useful identities

sin^{- 1} \frac{1}{x} = cosec^{- 1} x, cos^{- 1} \frac{1}{x} = sec^{- 1} x (∣ x ∣ \geq 1)

Useful identities

tan^{- 1} \frac{1}{x} = cot^{- 1} x (x > 0), - π + cot^{- 1} x (x < 0)

Useful identities

Worked examples

From the video

1. Domain and range of sin⁻¹(x²/2).

Domain [−√2, √2]; range [0, π/2].

2. cos⁻¹(−1/2)?

2π/3.

JEE-style question

Your turn

The value of sin⁻¹(sin(2π/3)) is:

(a)2π/3

(b)π/3

(c)−π/3

(d)π/6

Show the answer and the traps

2π/3 is outside [−π/2, π/2]. sin(2π/3) = sin(π/3), and π/3 is inside. Answer π/3: option (b).

(a) assumes sin⁻¹(sin x) = x for every x: the classic trap.

(c) gets the sign wrong: sin(2π/3) is positive. (d) is π/2 − π/3, as if the identity sin⁻¹x + cos⁻¹x = π/2 applied.

Watch out

Common mistakes

Writing sin⁻¹(sin x) = x for all xOnly on [−π/2, π/2].

Using tan⁻¹(1/x) = cot⁻¹x for negative xFor x < 0 it is −π + cot⁻¹x.

Giving cos⁻¹(−x) = −cos⁻¹xcos⁻¹(−x) = π − cos⁻¹x.

Practice

Try these

1. Find cos⁻¹(cos 7π/6).

5π/6.

2. Find tan⁻¹(−√3) + cot⁻¹(−1/√3).

−π/3 + 2π/3 = π/3.

3. Domain of sin⁻¹(2x − 1).

[0, 1].

4. If sin⁻¹x = π/5, find cos⁻¹x.

3π/10.