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

• General and Principal Values of sin$^{-1}$ x
• General and Principal Values of cos$^{-1}$ x
• General and Principal Values of tan$^{-1}$ x
• General and Principal Values of csc$^{-1}$ x
• General and Principal Values of sec$^{-1}$ x
• General and Principal Values of cot$^{-1}$ x
• Principal Values of Inverse Trigonometric Functions
• General Values of Inverse Trigonometric Functions
  
• arcsin(x) + arccos(x) = $\frac{π}{2}$
• arctan(x) + arccot(x) = $\frac{π}{2}$
• arctan(x) + arctan(y) = arctan($\frac{x + y}{1 - xy}$)
• arctan(x) - arctan(y) = arctan($\frac{x - y}{1 + xy}$)
• arctan(x) + arctan(y) + arctan(z)= arctan$\frac{x + y + z – xyz}{1 – xy – yz – zx}$
• arccot(x) + arccot(y) = arccot($\frac{xy - 1}{y + x}$)
• arccot(x) - arccot(y) = arccot($\frac{xy + 1}{y - x}$)
• arcsin(x) + arcsin(y) = arcsin(x $\sqrt{1 - y^{2}}$ + y$\sqrt{1 - x^{2}}$)
• arcsin (x) - arcsin(y) = arcsin (x $\sqrt{1 - y^{2}}$ - y$\sqrt{1 - x^{2}}$)
• arccos (x) + arccos(y) = arccos(xy - $\sqrt{1 - x^{2}}$$\sqrt{1 - y^{2}}$)
• arccos(x) - arccos(y) = arccos(xy + $\sqrt{1 - x^{2}}$$\sqrt{1 - y^{2}}$)
• 2 arcsin(x) = arcsin(2x$\sqrt{1 - x^{2}}$) 
• 2 arccos(x) = arccos(2x$^{2}$ - 1)
• 2 arctan(x) = arctan($\frac{2x}{1 - x^{2}}$) = arcsin($\frac{2x}{1 + x^{2}}$) = arccos($\frac{1 - x^{2}}{1 + x^{2}}$)
• 3 arcsin(x) = arcsin(3x - 4x$^{3}$)
• 3 arccos(x) = arccos(4x$^{3}$ - 3x)
• 3 arctan(x) = arctan($\frac{3x - x^{3}}{1 - 3 x^{2}}$)
• Inverse Trigonometric Function Formula
• Principal Values of Inverse Trigonometric Functions
• Problems on Inverse Trigonometric Function

11 and 12 Grade Math

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