Find the range of A=sin^2010 θ+cos^2014 θ

Name: Problem 59, Chapter 1: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-02-26T07:58:18-08:00
Duration: 1 min 34 s
Channel: Varsha Aggarwal
Description: Find the range of A=sin^2010 θ+cos^2014 θ

Key Concepts

Finding Extreme Values on a Closed Interval

Determining the range of a function involves identifying its maximum and minimum values on its domain. For functions composed of trigonometric terms raised to high even exponents, this typically entails analyzing the behavior at the endpoints as well as the critical points inside the interval. This approach is a standard part of studying functions and their ranges in mathematical analysis.

Substitution in Trigonometric Expressions

Introducing a substitution, such as letting t equal sin²? (or cos²?), can simplify expressions where sine and cosine appear with high even exponents. This technique reduces the complexity of the expression by transforming it into a function of a new variable that varies over a fixed interval, which facilitates further analysis of its range or optimization.

Use of Derivatives for Optimization

In order to locate the local extrema (minimum or maximum) within the interior of a function's domain, derivative tests are applied. By differentiating the function with respect to the substitution variable and setting the derivative equal to zero, one can solve for the potential critical points. This calculus method is crucial when determining the exact range of a function, especially when dealing with non-linear and high-powered expressions.

Even-Powered Trigonometric Functions

Raising sine and cosine functions to even exponents results in functions that are always nonnegative. This occurs because any real number raised to an even power yields a nonnegative result, regardless of the original sign of the sine or cosine value. This property is fundamental when considering the range of expressions involving even powers of trigonometric functions.

Range and Domain of Trigonometric Functions

The sine and cosine functions naturally oscillate between -1 and 1. When these functions are raised to even powers, their output is confined to the interval [0, 1]. Knowing the basic range of sine and cosine, as well as how powers transform these ranges, is essential for analyzing more complex expressions that are functions of sine and cosine.

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