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Inverse Trigonometric Functions — Class 12 Mathematics

Mathematics Part I · Chapter 2

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Chapter summary

Trigonometric functions are periodic and therefore not one-one over their full domains. To define inverses, each is restricted to a principal-value branch on which it is bijective. The chapter defines and their reciprocals, listing the principal ranges: , and . Several identities follow, such as and (for ). These functions and identities are essential tools for simplifying expressions, solving equations and integrating, where inverse trig functions appear as antiderivatives.

Principal value branchesInverse sine, cosine and tangentProperties and identitiesSimplification of inverse trig expressions

Key ideas & terms

Principal value branch: The restricted interval on which a trig function is bijective so its inverse is defined.
\(\sin^{-1}x\): Inverse sine, with range and domain .
\(\cos^{-1}x\): Inverse cosine, with range and domain .
\(\tan^{-1}x\): Inverse tangent, with range and domain .
Complementary identity: .
Addition formula: when .

Extra questions & answers

State the principal value range of .

The principal range is , with domain .

State the principal value range of .

The principal range is , with domain .

Evaluate .

Since and lies in the principal range, .

Prove .

Let ; then , so , giving the sum .

Evaluate .

and is in the principal range, so .

Simplify .

Since , use ; thus the value is .

State the value of .

For all real , .

Write the addition formula for .

provided .

Evaluate .

As , use ; hence the value is .

Why are trig functions restricted before inverting?

They are periodic and not one-one over their full domains; restricting to a principal branch makes them bijective so a single-valued inverse exists.