Inverse Trigonometric Functions Explained for Class 12 & Competitive Exams

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) = 1/2

2. 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^–1u

2. 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