ITF — 2. Domain & Range

2.1 What Are Domain and Range?

For any function, the domain is the set of all valid input values, and the range is the set of all possible output values.

For Inverse Trig Functions

The domain is the set of real numbers (trig ratio values) you can plug in. The range is the set of angles (in radians) that come out — restricted to the principal value branch.

2.2 Domain & Range of All Six Functions

Arcsine

sin⁻¹(x)

Domain[−1, 1]

Range[−π/2, π/2]

Degrees[−90°, 90°]

Arccosine

cos⁻¹(x)

Domain[−1, 1]

Range[0, π]

Degrees[0°, 180°]

Arctangent

tan⁻¹(x)

Domainℝ (all reals)

Range(−π/2, π/2)

Degrees(−90°, 90°)

Arccosecant

cosec⁻¹(x)

Domain|x| ≥ 1

Range[−π/2,π/2]\{0}

Degrees[−90°,90°]\{0°}

Arcsecant

sec⁻¹(x)

Domain|x| ≥ 1

Range[0,π]\{π/2}

Degrees[0°,180°]\{90°}

Arccotangent

cot⁻¹(x)

Domainℝ (all reals)

Range(0, π)

Degrees(0°, 180°)

2.3 Understanding the Ranges Visually

The ranges can be understood as positions on the number line between −π/2 and π. Closed brackets [ ] mean the endpoint is included; open brackets ( ) mean excluded.

sin⁻¹(x)

[−π/2, π/2] closed

cos⁻¹(x)

[0, π] closed

tan⁻¹(x)

(−π/2, π/2) open

cot⁻¹(x)

(0, π) open

Why are some ranges open?

For tan⁻¹, the endpoints ±π/2 correspond to vertical asymptotes of tan — tan is undefined there, so those angles cannot be outputs. Similarly for cot⁻¹ at 0 and π.

2.4 Complete Reference Table

Function	Domain	Range (Principal Branch)	Range in Degrees
sin⁻¹(x)	[−1, 1]	[−π/2, π/2]	[−90°, 90°]
cos⁻¹(x)	[−1, 1]	[0, π]	[0°, 180°]
tan⁻¹(x)	(−∞, +∞)	(−π/2, π/2)	(−90°, 90°)
cosec⁻¹(x)	(−∞,−1]∪[1,+∞)	[−π/2, π/2]\{0}	[−90°,90°]\{0°}
sec⁻¹(x)	(−∞,−1]∪[1,+∞)	[0, π]\{π/2}	[0°,180°]\{90°}
cot⁻¹(x)	(−∞, +∞)	(0, π)	(0°, 180°)

2.5 Key Observations

sin⁻¹ and cos⁻¹ share the same domain [−1, 1] (since sine and cosine both produce values between −1 and 1).
tan⁻¹ and cot⁻¹ accept all real numbers as input (since tan and cot can produce any real value).
cosec⁻¹ and sec⁻¹ exclude the interval (−1, 1) since cosec and sec always have |value| ≥ 1.
The range of sin⁻¹ and tan⁻¹ is symmetric around 0 — both are odd functions.
The range of cos⁻¹ starts at 0, not a negative number — cos⁻¹ is neither odd nor even.

⚠ Exam Trap

sin⁻¹(2) is undefined because 2 is outside the domain [−1, 1]. Always check the domain before evaluating!