Important Properties of Inverse Trigonometric Functions

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Properties of Inverse Trigonometric Functions Short Notes: Free PDF Download

Inverse trigonometric functions are periodic functions. This means a single trigonometric value x can satisfy many angles θ. To make these inverse trig functions that are both unique and invertible, We have to restrict the domain of the trigonometric functions (in other words range of the inverse trig function) where they are one-one and onto functions.

These restricted domains of trig ratios are called principal value branches, and they determine the range of the inverse trigonometric functions. Check the table below for all principal value branches.

Example:

If you calculate sin⁻¹(1/2), supposing it is equal to y;

y=sin⁻¹(1/2) = sin⁻¹(sin 30°)= 30° or π/6

This value π/6 or 30° lies within the principal value branch, which is [-π/2, π/2] or [ -90°, 90°].

Importance:

The concept of principal values is essential to make inverse trigonometric functions single-valued and invertible functions. There are an infinite number of general solutions for any inverse trig function, but only one unique solution that is the principal value.

You must be aware of domain and range of these inverse trigonometric functions.  Students must know the domains and ranges to solve numericals efficiency, and omit mistakes during the process.  Below table contain of the domain and range of all the Inverse trigonometric functions in class 11 Maths.

Students must know the basic properties of these inverse trigonometric functions. All these properties are essential to understanding the problems and solving them effectively.  Check below.

Negative Value

• sin⁻¹(-x) = -sin⁻¹(x) for x ∈ [-1, - {0} 
• cos⁻¹(-x) = π - cos⁻¹(x) for x ∈ [-1, - {0} 
• tan⁻¹(-x) = -tan⁻¹(x) if x > 0, or cot⁻¹(1/x) - π if x < 0 
• Cosec⁻¹(-x) = -Cosec⁻¹(x) for x ∈ (−∞,−1]
• sec⁻¹(-x) = π - sec⁻¹(x) for x ∈ (−∞,−1]
• cot⁻¹(-x) = -cot⁻¹(x) for x ∈ $(-\infty ,0)$

Reciprocal Value

• sin⁻¹(x) = cosec⁻¹(1/x) for x ∈ [-1, - {0} 
• cos⁻¹(x) = sec⁻¹(1/x) for x ∈ [-1, - {0} 
• tan⁻¹(x) = cot⁻¹(1/x) if x > 0, or cot⁻¹(1/x) - π if x < 0

Composition of Functions:

• sin(sin⁻¹(x)) = x for x ∈ [-1, 
• cos(cos⁻¹(x)) = x for x ∈ [-1, 
• tan(tan⁻¹(x)) = x for x ∈ R 
• sin⁻¹(sin(x)) = x for x ∈ [-π/2, π/2] 
• cos⁻¹(cos(x)) = x for x ∈ [0, π] 
• tan⁻¹(tan(x)) = x for x ∈ (-π/2, π/2) 

Check the important sum and difference formulas of CBSE and competitive exams.

• $${\sin }^{-1}(x)+{\sin }^{-1}(y)={\sin }^{-1}(x\sqrt{1-y^2}+y\sqrt{1-x^2})$$
• $$\sin^{-1}(x) + \sin^{-1}(y) = \sin^{-1}\left( x\sqrt{1 - y^2} + y\sqrt{1 - x^2} \right)$$
• $${\cos }^{-1}(x)+{\cos }^{-1}(y)={\cos }^{-1}(xy-\sqrt{(1-x^2)(1-y^2)})$$
• $${\cos }^{-1}(x)-{\cos }^{-1}(y)={\cos }^{-1}(xy+\sqrt{(1-x^2)(1-y^2)})$$
• $$\tan^{-1}(x) + \tan^{-1}(y) = \tan^{-1}\left( \frac{x + y}{1 - xy} \right)$$
• $$\tan^{-1}(x) - \tan^{-1}(y) = \tan^{-1}\left( \frac{x - y}{1 + xy} \right)$$
• $$\cot^{-1}(x) + \cot^{-1}(y) = \cot^{-1}\left( \frac{xy - 1}{x + y} \right)$$
• $${\cot }^{-1}(x)-{\cot }^{-1}(y)={\tan }^{-1}\left(\frac{y-x}{1+xy}\right)$$

Check other important properties of inverse trigonometry.

•   $si{n}^{-1}\left(x\right)+co{s}^{-1}\left(x\right)=\frac{\pi }{2}$ , $x\in \left[-1,1\right]$
•   $ta{n}^{-1}\left(x\right)+co{t}^{-1}\left(x\right)=\frac{\pi }{2}$ , $x\in R$
•   $cose{c}^{-1}\left(x\right)+se{c}^{-1}\left(x\right)=\frac{\pi }{2}$ , $\left|x\right|\ge 1$ i.e., $x\le -1$ or $x\ge 1$ i.e., $x\in R$ $-\left(-1,1\right)$

Double and Triple Angle Formulas 

• Numerically smallest angle is known as the principal value.
• The principal value is never numerically greater than π.
• Each inverse trigonometric function has a specific domain (the set of possible inputs) and range (the set of possible outputs). For instance, sin⁻¹ has a domain of [-1, 1] and a range of [-π/2, π/2].
• There exist many intervals other than $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$ where sine function is $1-1$ and therefore has inverse. But by $si{n}^{-1}x$ we shall always mean the function $si{n}^{-1}:\left[-1,1\right]$ $\to$ $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$ .
• Strictly monotone functions are $1-1$ and so invertible.
• $si{n}^{-1}x\ne {\left(sinx\right)}^{-1}$
• The principal value of $si{n}^{-1}x$ is the least numerical value among all the values of $si{n}^{-1}x$ . Principal value of $si{n}^{-1}x$ always belong to $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$ .
• If the branch of an inverse trigonometric function is not specifically mentioned, then we consider the principal branch only.
• Familiarize yourself with the principal values of inverse trigonometric functions for standard angles (0, π/6, π/4, π/3, π/2, etc.).
• Understand that sin(sin⁻¹(x)) = x for x in the domain and sin⁻¹(sin(x)) = x for x in the range of sin(x) (similar for other functions).
• Learn important identities like sin⁻¹(-x) = -sin⁻¹(x), cos⁻¹(-x) = π - cos⁻¹(x), tan⁻¹(-x) = -tan⁻¹(x), etc.
• Practice using the sum and difference identitiessuch as tan⁻¹(x) + tan⁻¹(y) = tan⁻¹((x + y) / (1 - xy)).