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 oneone 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 manyone 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 multiplevalued function; but a given value of θ gives a definite finite value of sin θ i.e., sin θ is a singlevalued 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 oneone 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 (cosinverse x) represents an angle and the cosine of this angle is equal to x. We have similar significances of the angles tan\(^{1}\)x (taninverse x), cot\(^{1}\)x (cotinverse x), sec\(^{1}\)x (secinverse x) and csc\(^{1}\)x (cscinverse 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.
11 and 12 Grade Math
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