Let f(x)=cos^-1(x) +cos^-1((x)/(2)+(√(3-3 x^2))/(2)), for (1)/(2) ≤x ≤1 Then find f(2013)

Name: Problem 142, Chapter 2: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-04-07T13:32:10-08:00
Duration: 1 min 23 s
Channel: Rukhmani Jain
Description: Let f(x)=cos^-1(x) +cos^-1((x)/(2)+(√(3-3 x^2))/(2)), for (1)/(2) ≤x ≤1 Then find f(2013)

step 1 In this question we are given the function that is f x is equal to cos inverse x plus inverse x

Let f(x)=cos^-1(x) +cos^-1((x)/(2)+(√(3-3 x^2))/(2)), for...

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as the inverse cosine (arccos), are used to retrieve an angle when its cosine value is known. They have specific domains and ranges that must be considered when solving problems, and their properties—including monotonic behavior and the particular intervals in which they are defined—are essential in understanding and manipulating expressions involving angles.

Trigonometric Identities

Trigonometric identities, like the cosine difference formula, play a crucial role in simplifying complex trigonometric expressions. These identities allow us to express a combination of sine and cosine terms as a single trigonometric function of a shifted angle, which can reveal hidden constants or simplify the structure of an equation. They are tools that provide deeper insight into the relationships between different trigonometric functions.

Function Constancy and Domain Considerations

Recognizing that a function may be constant over a specific domain often requires transforming the function using algebraic and trigonometric techniques to identify invariants. This understanding highlights that even when functions are defined on a restricted interval due to domain constraints—such as those imposed by an inverse trigonometric function—their intrinsic properties might extend beyond the initial definition under certain interpretations, leading to conclusions such as constant function values over broader inputs.

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