Inverse Trigonometric Functions
Inverse Trigonometric Functions are the inverse functions of the trigonometric ratios, i.e. sin, cos, tan, cot, sec, and cosec. These functions are widely used in fields like physics, mathematics, engineering, and other research fields. Just like addition and subtraction are the inverses of each other, the same is true for the inverse of trigonometric functions.
sin θ = x
⇒ θ = sin−1x
Representation of Inverse Trigonometric Functions
They are represented by adding arc in prefix or by adding -1 to the power.
Inverse sine can be written in two ways:
- sin-1 x
- arcsin x
Same goes for cos and tan.
Note: Don’t confuse sin-1 x with (sin x)-1. They are different. Writing sin-1 x is a way to write inverse sine whereas (sin x)-1 means 1/sin x.
Domain of Inverse Trigonometric Functions
We know that a function is differentiable only if it is continuous at that point and if a function is continuous at a given point then that point is the domain of the function. Hence we should learn the domain of the inverse trigonometric functions for the same.
Inverse Trigonometric Functions | Domain |
---|---|
sin-1x | [-1, 1] |
cos-1x | [-1, 1] |
tan-1x | R |
cosec-1x | (-∞, -1]∪[1, ∞) |
sec-1x | (-∞, -1]∪[1, ∞) |
cot-1x | R |
Now let’s learn the technique of implicit differentiation briefly.
Derivative of Inverse Trig Functions
Derivative of Inverse Trig Function refers to the rate of change in Inverse Trigonometric Functions. We know that the derivative of a function is the rate of change in a function with respect to the independent variable. Before learning this, one should know the formulas of differentiation of Trigonometric Functions. To find the derivative of the Inverse Trigonometric Function, we will first equate the trigonometric function with another variable to find its inverse and then differentiate it using the implicit differentiation formula.
In this article, we will learn the Derivative of Inverse Trig Functions, Formulas of Differentiation of Inverse Trig Functions, and Solve some Examples based on it. But before heading forward, let’s brush up on the concept of inverse trigonometric functions and implicit differentiation.
Table of Content
- Inverse Trigonometric Functions
- What is Implicit Differentiation?
- What is Derivative of Inverse Trigonometric Functions?
- Proof of Derivative of Inverse Trig Functions
- Inverse Trig Derivative Formula
- Inverse Trig Derivative Examples
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