medium difficulty8 min

Domain and Principal Value Branches of Inverse Trigonometric Functions

The principal value branch of an inverse trigonometric function is the chosen interval of angles on which the corresponding trigonometric function becomes one-one and onto its required range.

In one line: Inverse trig answers are not all angles; they are the one angle allowed by the function's selected branch.

Practice This Concept

Understand it

For an inverse trigonometric function to exist as a function, each input must give exactly one output. For example, sin θ = 1/2 has many angle solutions, but sin⁻¹(1/2) means the principal angle in [−π/2, π/2], which is π/6. The domain is the allowed input set, and the range is the selected principal value interval.

Example

sin⁻¹(−1/2) = −π/6 because −π/6 lies in [−π/2, π/2]. cos⁻¹(−1/2) = 2π/3 because 2π/3 lies in [0, π].

Formula and rule

Domains and ranges: sin⁻¹x: domain [−1,1], range [−π/2,π/2]; cos⁻¹x: domain [−1,1], range [0,π]; tan⁻¹x: domain R, range (−π/2,π/2); cot⁻¹x: domain R, range (0,π); sec⁻¹x: domain (−∞,−1] ∪ [1,∞), range [0,π] excluding π/2; cosec⁻¹x: domain (−∞,−1] ∪ [1,∞), range [−π/2,π/2] excluding 0.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Evaluate tan⁻¹(−1).

1Since tan⁻¹x has range (−π/2, π/2), the required angle θ must lie in this interval. tan θ = −1 gives θ = −π/4 in the principal range.
2Therefore tan⁻¹(−1) = −π/4.

Method you can reuse

Identify the inverse function, write its domain and principal range, check that the input is allowed, find an angle whose trigonometric value matches the input, then select only the angle lying in the principal range.

Visual tip

A branch diagram is useful: show the restricted interval on the x-axis for each trigonometric function and indicate that only this branch is inverted.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What is the principal value of cos⁻¹(−√3/2)?

Think first, then show answer

5π/6, because cos⁻¹x has range [0, π] and cos(5π/6) = −√3/2.

Common wrong answer

Using 3π/4 for tan⁻¹(−1) because tan(3π/4) = −1, even though 3π/4 is outside the range of tan⁻¹x.

Don't mix this up

Writing all possible angles instead of the principal value, such as giving sin⁻¹(1/2) = nπ + (−1)ⁿπ/6. That is a solution set for sin θ = 1/2, not the value of sin⁻¹(1/2).

Exam tip

Before evaluating, write the range of the inverse function beside it. This prevents choosing an angle from the wrong quadrant.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions usually ask for principal values, domains, ranges, or the validity of a simplification involving inverse functions.

1
Enough for 1 mark
The principal value branch of an inverse trigonometric function is the chosen interval of angles on which the corresponding trigonometric function becomes one-one and onto its required range.
2
Add for 2 marks
Example: sin⁻¹(−1/2) = −π/6 because −π/6 lies in [−π/2, π/2]. cos⁻¹(−1/2) = 2π/3 because 2π/3 lies in [0, π].
3
Add for 3 marks
For an inverse trigonometric function to exist as a function, each input must give exactly one output.

MCQ Quiz

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