C
CraftExam
medium difficulty8 min

Graphs of Inverse Trigonometric Functions

The graph of an inverse trigonometric function is obtained by reflecting the graph of the corresponding restricted trigonometric function in the line y = x.

In one line: Reflection swaps x and y, so domain and range also swap.

Practice This Concept

Understand it

A trigonometric graph must first be restricted to its principal branch so that it is one-one. After reflection in y = x, the old x-values become y-values and the old y-values become x-values. Thus the domain and range interchange.

Example

The graph of y = sin⁻¹x has domain [−1,1] and range [−π/2,π/2]. It is the reflection of y = sin x restricted to [−π/2,π/2] in the line y = x.

Formula and rule

Graph rule: if y = f⁻¹(x), then its graph is the reflection of y = f(x) in y = x, after f is restricted to a one-one principal branch.

Domain of f⁻¹ = range of restricted f, and range of f⁻¹ = domain of restricted f.

See it worked out

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

Worked example

Sketch y = sin⁻¹x.

  1. 1Start with y = sin x on [−π/2,π/2].
  2. 2Key points are (−π/2,−1), (0,0), (π/2,1).
  3. 3Reflect in y = x to get (−1,−π/2), (0,0), (1,π/2).
  4. 4Join these smoothly with increasing behaviour.
  5. 5Therefore the graph has domain [−1,1] and range [−π/2,π/2].

Method you can reuse

Choose the correct principal branch, list two or three key points on the restricted trigonometric graph, interchange coordinates, draw the reflected curve, and label the domain, range and endpoints.

Sketch it in your answer

Draw axes, the line y = x, the restricted original graph, and the reflected inverse graph. Label endpoints such as (−1,−π/2), (0,0), (1,π/2) for y = sin⁻¹x.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What are the domain and range of the graph y = cos⁻¹x?

Think first, then show answer
Domain [−1,1] and range [0,π].

Common wrong answer

Drawing a periodic inverse sine curve by copying the full sine wave shape after reflection.

Don't mix this up

Reflecting the full sine or cosine graph without restricting it first. This gives a curve that fails the vertical line test and is not a function.

Exam tip

When sketching, mark endpoints and the range interval first. A correct curve with missing endpoint labels can lose marks in graph questions.

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

Students may be asked to identify a graph, mark domain and range, label reflected points, or explain why a restricted branch is needed.
  1. 1

    Enough for 1 mark

    The graph of an inverse trigonometric function is obtained by reflecting the graph of the corresponding restricted trigonometric function in the line y = x.

  2. 2

    Add for 2 marks

    Example: The graph of y = sin⁻¹x has domain [−1,1] and range [−π/2,π/2]. It is the reflection of y = sin x restricted to [−π/2,π/2] in the line y = x.

  3. 3

    Add for 3 marks

    A trigonometric graph must first be restricted to its principal branch so that it is one-one.

MCQ Quiz

Practice this concept with focused MCQs

Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.

10 MCQs5 MinutesInstant Results
Practice This Concept

Help improve this page

Found something confusing, incorrect, or missing?

Practice This Concept