Definition formulas domain range and solved examples of inverse trigonometric functions
The concept of Inverse Trigonometric Functions plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios, especially when you need to find an angle from a given trigonometric value. This topic is vital in calculus, physics, engineering, and beyond. Let’s explore the essentials, applications, shortcuts, and common mistakes in a student-friendly format.
What Is Inverse Trigonometric Functions?
An Inverse Trigonometric Function is defined as the function that reverses the action of a basic trigonometric function, allowing us to calculate angles from known ratios such as sine, cosine, or tangent. You’ll find this concept applied in areas such as geometry, calculus, and real-life engineering problems like signal processing, navigation, and construction.
Standard Notation & Symbols
| Function Name | Symbol | Also Written As |
|---|---|---|
| Inverse Sine | sin–1x | arcsin x |
| Inverse Cosine | cos–1x | arccos x |
| Inverse Tangent | tan–1x | arctan x |
| Inverse Cotangent | cot–1x | arccot x |
| Inverse Secant | sec–1x | arcsec x |
| Inverse Cosecant | cosec–1x | arccosec x |
Domain and Range
| Function | Domain (x) | Range (y) | Principal Value |
|---|---|---|---|
| sin–1x | –1 ≤ x ≤ 1 | –π/2 ≤ y ≤ π/2 | [–π/2, π/2] |
| cos–1x | –1 ≤ x ≤ 1 | 0 ≤ y ≤ π | [0, π] |
| tan–1x | All real x | –π/2 < y < π/2 | (–π/2, π/2) |
| cot–1x | All real x | 0 < y < π | (0, π) |
| sec–1x | |x| ≥ 1 | 0 ≤ y ≤ π, y ≠ π/2 | [0, π]\{π/2} |
| cosec–1x | |x| ≥ 1 | –π/2 ≤ y ≤ π/2, y ≠ 0 | [–π/2, π/2]\{0} |
Formulas & Properties of Inverse Trigonometric Functions
Here are the most important formulas you’ll use in class, boards, and entrance exams:
- sin–1(–x) = –sin–1x
- cos–1(–x) = π – cos–1x
- tan–1(–x) = –tan–1x
- sin–1x + cos–1x = π/2
- tan–1x + cot–1x = π/2
- sec–1x + cosec–1x = π/2
For all formulas and a quick revision chart, see Trigonometric Identities (Class 10).
Graphs & Visualisation
The graph of each inverse trigonometric function is a reflection of its original function over the line y = x and is limited to its principal value range for uniqueness. For example, the graph of y = sin–1x exists only for x between –1 and 1. For visual reference, check out our Trigonometric Graphs page.
Solved Examples for Board & Competitive Exams
Example 1: Find the value of sin–1(1/2) + cos–1(1/2).
1. sin–1(1/2) = π/6 because sin(π/6) = 1/22. cos–1(1/2) = π/3 because cos(π/3) = 1/2
3. Their sum = π/6 + π/3 = π/2
Example 2: Differentiate y = tan–1(sin–12x) with respect to x.
1. Let u = sin–12x. Then y = tan–1u2. dy/dx = (1 / (1 + u2)) × (d/du of sin–12x with respect to x)
3. d/du of sin–12x = 2 / √(1 – 4x2)
4. Final Answer: dy/dx = [2 / (1 + (sin–12x)2)√(1 – 4x2)]
Integration & Derivatives
Here’s the standard differentiation and integration formulas for inverse trigonometric functions:
| Function | Derivative | Domain |
|---|---|---|
| sin–1x | 1 / √(1 – x2) | |x| < 1 |
| cos–1x | –1 / √(1 – x2) | |x| < 1 |
| tan–1x | 1 / (1 + x2) | All real x |
| cot–1x | –1 / (1 + x2) | All real x |
| sec–1x | 1 / (|x|√(x2 – 1)) | |x| > 1 |
| cosec–1x | –1 / (|x|√(x2 – 1)) | |x| > 1 |
For detailed explanation and solved calculus problems, visit our Derivatives page.
Speed Trick or Vedic Shortcut
A common trick for remembering domains and principal value ranges is to use a color-coded chart or hand mnemonic. For example, remember: “sin–1” and “tan–1” range from –π/2 to π/2 (think S and T share a symmetric range about zero), while “cos–1” is from 0 to π (C for 'Closed' in the top half-circle). These memory pegs save time in competitive exams like JEE and NEET. Explore more tricks with Vedantu’s live classes for instant recall tips.
Try These Yourself
- Find the principal value of cos–1(–1/2).
- Evaluate tan–1(1) + cot–1(–1).
- Differentiate y = sin–1(2x).
- What is the domain and range of sec–1x?
Frequent Errors and Misunderstandings
- Forgetting to consider the proper domain/range for input values.
- Mistaking sin–1x for (sin x)–1 (not the same: sine reciprocal vs. inverse!).
- Leaving out the absolute value in derivatives of sec–1x and cosec–1x.
- Missing negative signs in formulas of cos–1x, cot–1x, and cosec–1x derivatives.
Relation to Other Concepts
The idea of Inverse Trigonometric Functions connects closely with topics such as Trigonometric Functions. Mastering these helps with integration techniques, proving identities, and solving equations in higher mathematics and physics.
Classroom Tip
A quick way to remember inverse trig formula sets is to keep a handwritten chart or sticky note on your study table. Vedantu’s expert educators recommend filling in blank formula tables daily for a week before exams!
We explored Inverse Trigonometric Functions—from definition, notations, formulas, and common mistakes, to tried-and-tested tricks. Keep practicing with more solved problems on Vedantu to master this important topic for all your exams and real-world applications!
Related Links: Trigonometric Functions, Integration by Parts Rule
FAQs on Inverse Trigonometric Functions Explained for Students
1. What are inverse trigonometric functions?
Inverse trigonometric functions are functions that give the angle whose trigonometric ratio is known. They are the inverses of sine, cosine, tangent, cosecant, secant, and cotangent.
- The main inverse functions are sin⁻¹x (arcsin x), cos⁻¹x (arccos x), and tan⁻¹x (arctan x).
- If sin θ = x, then θ = sin⁻¹x.
- They are also called arc functions because they return angles.
2. What is the domain and range of inverse trigonometric functions?
The domain and range of inverse trigonometric functions are restricted so they become one-to-one and have proper inverses.
- sin⁻¹x: Domain = [-1, 1], Range = [-π/2, π/2]
- cos⁻¹x: Domain = [-1, 1], Range = [0, π]
- tan⁻¹x: Domain = (-∞, ∞), Range = (-π/2, π/2)
3. What is the principal value in inverse trigonometric functions?
The principal value is the unique angle selected from the restricted range of an inverse trigonometric function.
- Since trigonometric functions are periodic, they have multiple angles for the same value.
- To make them invertible, we restrict their range.
- For example, sin⁻¹(1/2) = π/6, not 5π/6.
4. How do you evaluate sin⁻¹, cos⁻¹, and tan⁻¹ values?
To evaluate inverse trigonometric functions, find the angle in the principal range whose trigonometric ratio equals the given value.
- sin⁻¹(√3/2) = π/3 because sin(π/3) = √3/2.
- cos⁻¹(1/2) = π/3 because cos(π/3) = 1/2.
- tan⁻¹(1) = π/4 because tan(π/4) = 1.
5. What is the formula for tan⁻¹x + cot⁻¹x?
The identity for tan⁻¹x and cot⁻¹x is tan⁻¹x + cot⁻¹x = π/2 (for all real x).
- This identity is valid when both functions are defined using principal values.
- Example: If x = 1, then tan⁻¹1 = π/4 and cot⁻¹1 = π/4.
- Their sum is π/4 + π/4 = π/2.
6. What is the derivative of inverse trigonometric functions?
The derivatives of inverse trigonometric functions follow standard formulas used in calculus.
- d/dx (sin⁻¹x) = 1/√(1 − x²)
- d/dx (cos⁻¹x) = −1/√(1 − x²)
- d/dx (tan⁻¹x) = 1/(1 + x²)
7. How do you solve equations involving inverse trigonometric functions?
To solve equations with inverse trigonometric functions, convert them into basic trigonometric form and use known values.
- Example: Solve sin⁻¹x = π/6.
- Take sine on both sides: x = sin(π/6).
- Since sin(π/6) = 1/2, the solution is x = 1/2.
8. What is the difference between sin⁻¹x and 1/sin x?
The expression sin⁻¹x means inverse sine (arcsin), while 1/sin x means cosecant (csc x).
- sin⁻¹x gives an angle whose sine is x.
- 1/sin x = csc x is a reciprocal trigonometric function.
- They are completely different operations.
9. What are some important identities of inverse trigonometric functions?
Important identities help simplify expressions involving inverse trigonometric functions.
- sin⁻¹x + cos⁻¹x = π/2
- tan⁻¹x + tan⁻¹(1/x) = π/2 (for x > 0)
- sin⁻¹(−x) = −sin⁻¹x
10. Where are inverse trigonometric functions used in real life?
Inverse trigonometric functions are used to calculate angles from known ratios in real-world applications.
- In physics to determine angles in projectile motion.
- In engineering and navigation to find direction and elevation angles.
- In computer graphics for rotation and orientation calculations.
