Inverse of Cosine Function, y = cos-1(x)
cos-1(x) is the inverse function of cos(x). Its domain is [−1, 1] and its range is [0, π]. It intersects the coordinate axis at (1, π/2). It is neither even nor an odd function and is strictly decreasing in (-1, 1).
Graph of Function
Function Analysis
Domain | [Tex]x ∈ [−1, 1][/Tex] |
---|---|
Range | [Tex]y ∈ [0, \pi][/Tex] |
X – Intercept | [Tex]x = 1[/Tex] |
Y – Intercept | [Tex]y = \frac{\pi}{2}[/Tex] |
Minima | [Tex](1, 0)[/Tex] |
Maxima | [Tex](-1, \pi)[/Tex] |
Inflection Points | [Tex]\big(0, \frac{\pi}{2}\big)[/Tex] |
Parity | Neither Even Nor Odd |
Monotonicity | In (-1, 1) strictly decreasing |
Sample Problems on Inverse Cosine Function
Problem 1: Find the principal value of the given equation:
y = cos-1(1/√2)
Solution:
We are given that:
y = cos-1(1/√2)
So we can say that,
cos(y) = (1/√2)
We know that the range of the principal value branch of cos-1(x) is (0, π) and cos(π/4) = 1/√2.
So, the principal value of cos-1(1/√2) = π/4.
Problem 2: Find the principal value of the given equation:
y = cos-1(1)
Solution:
We are given that:
y = cos-1(1)
So we can say that,
cos(y) = 1
We know that the range of the principal value branch of cos-1(x) is (0, π) and cos(0) = 1.
So, the principal value of cos-1(1) = 0.
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)