# Inverse Trigonometric Functions

We will discuss here about Inverse trigonometric Functions or inverse circular functions.

The inverse of a function f: A B exists if and only if f is one-one onto (i.e., bijection) and given by

f(x) = y⇔ f$$^{-1}$$ (y) = x.

Consider the sine function. Clearly, sin: R  R given by sin θ = x for all θ ∈ R is a many-one into function. So, its inverse does not exist. If we restrict its domain to the interval [- $$\frac{π}{2}$$, $$\frac{π}{2}$$] then we may have infinitely many values of the angle θ which satisfy the equation sin θ = x i.e., sine of any one of these angles is equal to x. Here angle θ is represented as sin$$^{-1}$$x which is read as sine inverse x or arc sin x. Therefore, the symbol sin$$^{-1}$$x represents an angle and the sine of this angle has the value x.

Note the difference between sin$$^{-1}$$x and sin θ: sin$$^{-1}$$x represents an angle while sin θ represents a pure number; again, for a given value of x (- 1 ≤ x ≤ 1) we may have infinitely many vales of sin$$^{-1}$$x i.e., sin$$^{-1}$$x is a multiple-valued function; but a given value of θ gives a definite finite value of sin θ i.e., sin θ is a single-valued function. Thus, if x is a real number lying between -1 and 1, then sin$$^{-1}$$ x is an angle between - $$\frac{π}{2}$$ and $$\frac{π}{2}$$ whose sine is x i.e.,

sin$$^{-1}$$x = θ

⇔ x = sin θ, where - $$\frac{π}{2}$$  ≤ x ≤ $$\frac{π}{2}$$ and - 1 ≤ x ≤ 1.

In the above discussion we have restricted the sine function to the interval [- $$\frac{π}{2}$$, $$\frac{π}{2}$$] to ake it a bijection. In fact we restrict the domain of sin θ to any of the interval [- $$\frac{π}{2}$$, $$\frac{π}{2}$$], [$$\frac{3π}{2}$$, $$\frac{5π}{2}$$], [- $$\frac{5π}{2}$$, -$$\frac{3π}{2}$$] etc. sin θ is one-one onto function with range [-1, 1]. We therefore conclude that each of these intervals we can define the inverse of sine function. Thus sin$$^{-1}$$x is a function with domain [-1, 1] = {x ∈ R: - 1 ≤ x ≤ 1} and range [- $$\frac{π}{2}$$, $$\frac{π}{2}$$] or [$$\frac{3π}{2}$$, $$\frac{5π}{2}$$] or [- $$\frac{5π}{2}$$, -$$\frac{3π}{2}$$] and so on.

Similarly, if cos θ = x (- 1 ≤ x ≤ 1 ) then θ = cos$$^{-1}$$x i.e., cos$$^{-1}$$x (cos-inverse x) represents an angle and the cosine of this angle is equal to x. We have similar significances of the angles tan$$^{-1}$$x (tan-inverse x), cot$$^{-1}$$x (cot-inverse x), sec$$^{-1}$$x (sec-inverse x) and csc$$^{-1}$$x (csc-inverse x).

Therefore, if sin θ = x (- 1 ≤ x ≤ 1) then θ = sin$$^{-1}$$x;

if cos θ = x (- 1 ≤ x ≤ 1) then θ = cos$$^{-1}$$x ;

if tan θ = x (- ∞ < x < ∞) then θ = tan$$^{-1}$$x ;

if csc θ = x (I x I ≥ 1) then θ = csc$$^{-1}$$x.

if sec θ = x (I x I ≥ 1) then θ = sec$$^{-1}$$x ; and

if cot θ = x (- ∞ < x < ∞) then θ = cot$$^{-1}$$x ;

Conversely, sin$$^{-1}$$x = θ ⇒ sin θ = x;

cos$$^{-1}$$x = θ ⇒ cos θ = x

tan$$^{-1}$$x = θ ⇒ tan θ = x

csc$$^{-1}$$x = θ ⇒ csc θ = x

cot$$^{-1}$$x = θ ⇒ cot θ = x

The trigonometrical functions sin$$^{-1}$$x, cos$$^{-1}$$x, tan$$^{-1}$$x, cot$$^{-1}$$x, sec$$^{-1}$$x and csc$$^{-1}$$x are called Inverse Circular Functions.

Note: It should be noted that sin$$^{-1}$$x is not equal to (sin x)$$^{-1}$$. Also noted that (sin x)$$^{-1}$$is an angle whose sin is x. Remember that sin$$^{-1}$$x is a circular function but (sin x )$$^{-1}$$ is the reciprocal of sin x i.e., (sin x)$$^{-1}$$ = 1/sin x and it represents a pure number.

Inverse Trigonometric Functions