Inverse of Tangent Function, y = tan-1(x)
tan-1(x) is the inverse function of tan(x). Its domain is ℝ and its range is [-π/2, π/2]. It intersects the coordinate axis at (0, 0). It is an odd function which is strictly increasing in (-∞, ∞).
Graph of Function
Function Analysis
| Domain | [Tex]x ∈ ℝ[/Tex] |
|---|---|
| Range | [Tex]y ∈ \big(\frac{-\pi}{2}, \frac{\pi}{2}\big)[/Tex] |
| X – Intercept | [Tex]x = 0[/Tex] |
| Y – Intercept | [Tex]y = 0[/Tex] |
| Minima | The function does not have any minima points. |
| Maxima | The function does not have any maxima points. |
| Inflection Points | [Tex](0, 0)[/Tex] |
| Parity | Odd Function |
| Monotonicity | In (−∞, ∞) strictly Increasing |
| Asymptotes | [Tex]y = \frac{\pi}{2}\space and\space y= \frac{-\pi}{2}[/Tex] |
Sample Problems on Inverse of Tangent Function
Problem 1: Find the principal value of the given equation:
y = tan-1(1)
Solution:
We are given that:
y = tan-1(1)
So we can say that,
tan(y) = (1)
We know that the range of the principal value branch of tan-1(x) is (-π/2, π/2) and tan(π/4) = 1.
So, the principal value of tan-1(1) = π/4.
Problem 2: Find the principal value of the given equation:
y = tan-1(√3)
Solution:
We are given that:
y = tan-1(√3)
So we can say that,
tan(y) = (√3)
We know that the range of the principal value branch of tan-1(x) is (-π/2, π/2) and tan(π/3) = √3.
So, the principal value of tan-1(√3) = π/3.
Graphs of Inverse Trigonometric Functions – Trigonometry | Class 12 Maths
Inverse trigonometric functions are the inverse functions of the trigonometric ratios i.e. sin, cos, tan, cot, sec, cosec. These functions are widely used in fields like physics, mathematics, engineering and other research fields. There are two popular notations used for inverse trigonometric functions:
Adding “arc” as a prefix.
Example: arcsin(x), arccos(x), arctan(x), …
Adding “-1” as superscript.
Example: sin-1(x), cos-1(x), tan-1(x), …
In this article, we will learn about graphs and nature of various inverse functions.
Table of Content
- Inverse of Sine Function, y = sin-1(x)
- Inverse of Cosine Function, y = cos-1(x)
- Inverse of Tangent Function, y = tan-1(x)
- Inverse of Cosecant Function, y = cosec-1(x)
- Inverse of Secant Function, y = sec-1(x)
- Inverse of Cotangent Function, y = cot-1(x)
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