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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    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

    Then show how to use it

    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.

  3. 3

    Add one concrete 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.

  4. 4

    Avoid this incomplete answer

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

Definition

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

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.

Rule to remember

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.

Memory hook

Reflection swaps x and y, so domain and range also swap.

Examples and method

Worked example

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

Method to apply

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.

Diagram support

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.

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.

Avoid common mistakes

Common confusion

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.

Common wrong answer

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

Exam tip

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

Quick check

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

Domain [−1,1] and range [0,π].

Answer writing and exam use

1-mark answer

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-mark answer

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

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. 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. Sketch y = sin⁻¹x. Start with y = sin x on [−π/2,π/2]. Key points are (−π/2,−1), (0,0), (π/2,1). Reflect in y = x to get (−1,−π/2), (0,0), (1,π/2). Join these smoothly with increasing behaviour. Therefore the graph has domain [−1,1] and range [−π/2,π/2]. Students may be asked to identify a graph, mark domain and range, label reflected points, or explain why a restricted branch is needed. Drawing a periodic inverse sine curve by copying the full sine wave shape after reflection.
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