How to Draw and Understand Graphs of Inverse Trigonometric Functions with Domain and Range
The concept of graphic representation of inverse trigonometric function is essential in mathematics and helps students easily understand how these special functions behave through their graphs, domains, and ranges. These visualizations are heavily used in board exams, JEE, and other competitive tests.
Understanding Graphic Representation of Inverse Trigonometric Function
A graphic representation of inverse trigonometric function visually displays the behavior, domain, range, and properties of the inverse trigonometric (arc) functions such as arcsin, arccos, and arctan. This concept is widely used in calculus applications, exam revision, and solving trigonometric equations. It helps explain domain and range relations, recognizes symmetry, and clarifies principal value branches for each inverse function.
Key Concepts: Domain, Range, and Principal Branches
Knowing the domains and ranges of each inverse trigonometric function is the first step before plotting their graphs. Only the principal value branch (the range where the function is single valued) is considered for each inverse. Here is a summary:
| Function | Domain | Principal Value Range |
|---|---|---|
| sin⁻¹x (arcsin) | -1 ≤ x ≤ 1 | -π/2 ≤ y ≤ π/2 |
| cos⁻¹x (arccos) | -1 ≤ x ≤ 1 | 0 ≤ y ≤ π |
| tan⁻¹x (arctan) | All Real x | -π/2 < y < π/2 |
| cot⁻¹x (arccot) | All Real x | 0 < y < π |
| sec⁻¹x (arcsec) | x ≤ -1 or x ≥ 1 | 0 ≤ y ≤ π, y ≠ π/2 |
| cosec⁻¹x (arccosec) | x ≤ -1 or x ≥ 1 | -π/2 ≤ y ≤ π/2, y ≠ 0 |
This table shows the relationship of the function, its allowed input values, and where its inverse is uniquely defined.
Step-by-Step: How to Draw the Graph of sin⁻¹x, cos⁻¹x, and tan⁻¹x
Let’s plot the most common inverse trigonometric functions one by one:
How to graph sin⁻¹x (arcsin x):
1. Start with the original y = sinx graph, restrict its domain to [-π/2, π/2] where sinx is one-to-one.
2. Swap x and y axes to get the inverse graph (reflect y = sinx in y = x).
3. The resulting graph is defined only for x in [-1, 1].
4. Draw a smooth increasing curve from (-1, -π/2) passing through (0,0) and up to (1, π/2).
How to graph cos⁻¹x (arccos x):
1. Take the portion of y = cosx for x in [0, π].
2. Interchange x and y to reflect about y = x.
3. The curve is defined for x in [-1, 1].
4. Draw a decreasing curve from (-1, π) to (1, 0).
How to graph tan⁻¹x (arctan x):
1. Use y = tanx for -π/2 < x < π/2 (where tanx is one-to-one).
2. Swap axes (reflect about y = x) to get the inverse.
3. The graph is defined for all real x.
4. Draw an increasing S-shaped curve approaching y = π/2 as x → ∞ and y = -π/2 as x → -∞, passing through (0,0).
Graph Properties and Behaviors
- All standard graphic representations of inverse trigonometric function are smooth and continuous.
- arcsin(x) and arctan(x) are increasing functions, while arccos(x) is decreasing in its domain.
- Important symmetry: arcsin(-x) = -arcsin(x) and arctan(-x) = -arctan(x) (odd functions).
- Intercepts: All graphs pass through (0,0), except arccos, which passes through (1,0).
- The endpoints of arcsin and arccos at x = ±1 are crucial to label in exams.
Formulae Used in Graphic Representation of Inverse Trigonometric Function
The standard formulas for inverse trigonometric functions help in drawing and solving problems:
sin(sin⁻¹x) = x for -1 ≤ x ≤ 1
cos(cos⁻¹x) = x for -1 ≤ x ≤ 1
tan(tan⁻¹x) = x for all real x
These ensure students know when to use each graph and avoid domain/range errors.
Worked Example – Plotting sin⁻¹x Graph
1. Choose key points: x = -1, 0, 1
2. Calculate y-values: sin⁻¹(-1) = -π/2, sin⁻¹(0) = 0, sin⁻¹(1) = π/2
3. Mark (-1, -π/2), (0, 0), (1, π/2) on axes.
4. Draw a smooth increasing curve passing through these points within the box x ∈ [-1, 1], y ∈ [−π/2, π/2].
Common Mistakes to Avoid
- Confusing the domain and range positions for each graph.
- Forgetting principal value branches and plotting multiple cycles.
- Not labeling endpoints such as (1, π/2) on arcsin or (-1, π) on arccos.
- Mixing the graph shapes of arcsin, arccos, and arctan.
Real-World and Exam Applications
The graphic representation of inverse trigonometric function plays a vital role in calculus (finding integrals), engineering, and even navigation. In exams, students must sketch accurate graphs to answer MCQs, evaluate limits, or solve trigonometric equations. Practicing graphing with resources from Vedantu strengthens conceptual understanding.
Page Summary
We explored the graphic representation of inverse trigonometric function, their domains, ranges, stepwise graph plotting, and practical usage. Consistent practice, clear graph labeling, and understanding of principal branches help students score high in exams and tackle advanced trigonometry confidently. Learn more and practice similar concepts with Vedantu for exam success.
Related Topics and Further Learning
- Inverse Trigonometric Functions
- Trigonometric Functions
- Domain and Range Relations
- Properties of Inverse Trigonometric Functions
- Introduction to Trigonometry
- Application of Trigonometry
- Graphs and Graphical Representation
- Graphical Representation of Data
- Sinx Graph
FAQs on Graphic Representation of Inverse Trigonometric Functions Explained Clearly
1. What is the graph of an inverse trigonometric function?
The graph of an inverse trigonometric function is the reflection of the corresponding trigonometric function across the line y = x, restricted to a suitable domain so it becomes one-to-one.
- Inverse functions include arcsin x, arccos x, arctan x, etc.
- Because trigonometric functions are periodic, their domains are restricted before finding the inverse.
- The graph is obtained by swapping x and y coordinates of the original restricted function.
2. How do you draw the graph of y = arcsin x?
The graph of y = arcsin x is drawn by reflecting the restricted graph of y = sin x (for −π/2 ≤ x ≤ π/2) across y = x.
- Step 1: Draw y = sin x for −π/2 ≤ x ≤ π/2.
- Step 2: Reflect the curve across the line y = x.
- Domain of arcsin x: −1 ≤ x ≤ 1.
- Range of arcsin x: −π/2 ≤ y ≤ π/2.
3. What is the domain and range of inverse trigonometric functions?
The domain of an inverse trigonometric function is the range of its corresponding restricted trig function, and its range is the restricted domain.
- arcsin x: Domain = [−1, 1], Range = [−π/2, π/2]
- arccos x: Domain = [−1, 1], Range = [0, π]
- arctan x: Domain = (−∞, ∞), Range = (−π/2, π/2)
4. Why do we restrict the domain when graphing inverse trigonometric functions?
We restrict the domain because trigonometric functions are not one-to-one, and an inverse exists only for one-to-one functions.
- Functions like sin x and cos x are periodic.
- Without restriction, they fail the horizontal line test.
- Restricting the domain ensures each output corresponds to exactly one input.
5. What is the graph of y = arctan x?
The graph of y = arctan x is an increasing S-shaped curve with horizontal asymptotes at y = −π/2 and y = π/2.
- Domain: (−∞, ∞)
- Range: (−π/2, π/2)
- It passes through the point (0, 0).
- It is the reflection of y = tan x (restricted to −π/2 < x < π/2) across y = x.
6. How is the graph of arccos x different from arcsin x?
The main difference between arccos x and arcsin x graphs lies in their ranges.
- arcsin x has range [−π/2, π/2] and is increasing.
- arccos x has range [0, π] and is decreasing.
- Both have domain [−1, 1].
- Their shapes are reflections of restricted sine and cosine graphs across y = x.
7. What are the asymptotes of inverse trigonometric graphs?
The inverse trigonometric functions arctan x and arccot x have horizontal asymptotes.
- For arctan x, asymptotes are y = −π/2 and y = π/2.
- For arccot x, asymptotes are y = 0 and y = π (depending on definition).
- arcsin x and arccos x do not have asymptotes because their domains are limited.
8. Can you give an example of plotting an inverse trigonometric function?
To plot y = arcsin x, use key points from the sine function and reflect them across y = x.
- sin(−π/2) = −1 → arcsin(−1) = −π/2
- sin(0) = 0 → arcsin(0) = 0
- sin(π/2) = 1 → arcsin(1) = π/2
- Plot points (−1, −π/2), (0, 0), (1, π/2) and draw a smooth increasing curve.
9. What is the relationship between a trigonometric function and its inverse graph?
The graph of an inverse trigonometric function is the reflection of the restricted trigonometric graph across the line y = x.
- If (a, b) lies on y = sin x, then (b, a) lies on y = arcsin x.
- This reflection property applies to all inverse functions.
- The restriction ensures the inverse passes the horizontal line test.
10. What are common mistakes when graphing inverse trigonometric functions?
A common mistake when graphing inverse trigonometric functions is forgetting to restrict the domain before reflecting the graph.
- Not applying the correct principal value range.
- Confusing domain and range after inversion.
- Drawing the full periodic trig graph instead of the restricted portion.
- Ignoring asymptotes for functions like arctan x.
