Basics Concepts: Inverse Trigonometric Functions | Mathematics (Maths) for JEE Main & Advanced PDF Download

What is an Inverse Function?

A function accepts an input, performs a rule, and produces an output. An inverse function reverses this process: it accepts the output and returns the original input.

• If y = f(x) and x = g(y) are two functions such that f(g(y)) = y for all y in the domain of g and g(f(x)) = x for all x in the domain of f, then f and g are inverses of each other.
• Example: If f(x) = 2x + 5 = y, then g(y) = (y - 5)/2 = x is the inverse of f. The inverse of f is denoted by f^-1.

Inverse Trigonometric Functions

Ordinary trigonometric functions (sin, cos, tan, cot, sec, cosec) are not one-to-one on their entire natural domains; they are many-to-one. Hence an inverse in the usual sense does not exist unless we restrict the domain so that the function becomes one-to-one (injective). On such a restricted domain we can define a unique inverse called the principal inverse.

• For example, y = sin x is many-to-one on R. If we restrict sin x to the interval [-π/2, π/2], it becomes one-to-one and we define the principal inverse sin^-1 (also written arcsin) with range [-π/2, π/2].
• Similarly we define principal values for cos^-1 (arccos), tan^-1 (arctan), cot^-1, sec^-1 and cosec^-1 on suitable restricted domains.

Principal-value notation and multivaluedness

For a trigonometric equation z = sin ω there are infinitely many ω giving the same z. The notation sin^-1z usually denotes the principal value, the unique value chosen from a specified range (for arcsin it is [-π/2, π/2]). Some texts write arcsin z to emphasise the principal value.

Principal-value domains (summary)

The principal branches (domain of the inverse function and range of the original) are normally taken as follows.

Fig: Inverse Trigonometric Functions

Basic identities and remarks

• For x in the domain of the inverse, sin(sin^-1x) = x and cos(cos^-1x) = x, etc.
• For angles y in the principal range, sin^-1(sin y) = y when y lies in [-π/2, π/2]. For y outside the principal range, sin^-1(sin y) gives the principal value equivalent to y.
• Inverse trigonometric functions are useful to recover angles from known trigonometric ratios and appear frequently in geometry, calculus and applied problems.

Solved Examples

Ques 1: Find the exact value of each expression without a calculator, in [0, 2π).

1. sin^-1(-√3/2)
2. cos^-1(-√2/2)
3. tan^-1(√3)

Ans:

• For sin^-1(-√3/2):

sin θ = -√3/2. The reference angle with sine √3/2 is π/3.

The value of θ in the principal range [-π/2, π/2] where sine is -√3/2 is θ = -π/3.

Therefore sin^-1(-√3/2) = -π/3.

• For cos^-1(-√2/2):

cos θ = -√2/2. The reference angle with cosine √2/2 is π/4.

The principal value of arccos lies in [0, π] where cosine is negative in (π/2, π). The angle is θ = 3π/4.

Therefore cos^-1(-√2/2) = 3π/4.

• For tan^-1(√3):

tan θ = √3. The reference angle is π/3.

The principal value of arctan lies in (-π/2, π/2), so θ = π/3.

Therefore tan^-1(√3) = π/3.

Note. If the problem asked for all solutions in [0, 2π), each trig equation has multiple solutions (for example sin θ = -√3/2 gives θ = 4π/3 and 5π/3 in [0,2π)). But inverse functions by convention return the principal value.

Ques 2: Find the value of tan^-1(1.1106).

Ans:

A = tan^-1(1.1106)

tan A = 1.1106

Using a calculator in degree mode gives A ≈ 48°.

Therefore tan^-1(1.1106) ≈ 48° (or in radians ≈ 0.8378 rad).

Properties of Inverse Trigonometric Functions

We list commonly used properties and give proofs or short justifications. In each identity the domains indicated ensure expressions are defined.

Property 1 - Reciprocal relations

1. sin^-1(1/x) = cosec^-1x, for |x| ≥ 1
2. cos^-1(1/x) = sec^-1x, for |x| ≥ 1
3. tan^-1(1/x) = cot^-1x, for x ≠ 0; a sign convention is used to select principal values (commonly true for x > 0 with standard principal branches)

Proof (first relation) - sin^-1(1/x) = cosec^-1x for |x| ≥ 1:

Let y = sin^-1(1/x).

Then sin y = 1/x and so cosec y = x, provided x ≠ 0.

Thus y = cosec^-1x (with the usual principal-value choices), so sin^-1(1/x) = cosec^-1x.

Property 2 - Oddness of certain inverses

1. sin^-1(-x) = -sin^-1(x), x ∈ [-1, 1]
2. tan^-1(-x) = -tan^-1(x), x ∈ R
3. cosec^-1(-x) = -cosec^-1(x), |x| ≥ 1

Proof - sin^-1(-x) = -sin^-1(x):

Let y = sin^-1(-x).

Then sin y = -x.

So sin(-y) = -sin y = x.

Therefore sin^-1(x) = -y, which gives y = -sin^-1(x).

Hence sin^-1(-x) = -sin^-1(x) for x ∈ [-1, 1].

Property 3 - Relations with π

1. cos^-1(-x) = π - cos^-1(x), x ∈ [-1, 1]
2. sec^-1(-x) = π - sec^-1(x), |x| ≥ 1
3. cot^-1(-x) = π - cot^-1(x), x ∈ R (with principal branch (0, π))

Proof - cos^-1(-x) = π - cos^-1(x):

Let y = cos^-1(-x).

Then cos y = -x, so -x = cos y = cos(π - y).

Thus cos(π - y) = x, so cos^-1(x) = π - y.

Therefore y = π - cos^-1(x), i.e. cos^-1(-x) = π - cos^-1(x).

More Worked Examples

Ques 1: Prove that sin^-1(-x) = -sin^-1(x), x ∈ [-1,1]

Ans:

Assume y = sin^-1(-x).

Then sin y = -x.

So sin(-y) = -sin y = x.

Therefore sin^-1(x) = -y.

Hence y = -sin^-1(x), which demonstrates sin^-1(-x) = -sin^-1(x).

Ques 2: sin^-1(cos π/3) = ?

Ans:

cos(π/3) = 1/2.

sin^-1(1/2) = π/6 (principal value in [-π/2, π/2]).

Therefore sin^-1(cos π/3) = π/6.

Ques 3: Find the value of sin(π/4 + cos^-1(√2/2)).

Ans:

Let A = cos^-1(√2/2).

Then cos A = √2/2 = 1/√2.

The principal value A in [0, π] satisfying cos A = 1/√2 is A = π/4.

Therefore sin(π/4 + cos^-1(√2/2)) = sin(π/4 + π/4) = sin(π/2) = 1.

Remarks and Common Exam Tips

• Always check the principal range before giving the value of an inverse trig function; inverse notation denotes the principal value, not all possible angles.
• Remember the standard values for sin, cos and tan at 0, π/6, π/4, π/3, π/2 and their negatives; these are used extensively.
• For expressions containing inverse trig functions and algebraic manipulations, draw a right triangle or use identities such as sin²θ + cos²θ = 1 to simplify.
• Watch out for domain restrictions: sin^-1 and cos^-1 accept only inputs in [-1, 1]; sec^-1 and cosec^-1 accept only |x| ≥ 1.

This completes the discussion on the basic concepts of inverse trigonometric functions, their principal values, domains and ranges, standard properties, and worked examples useful for school-level and competitive examinations.