Find the minimum value of f(θ)=(3 sin(θ)-4 cos(θ)-10) (3 sin(θ)+4 cos(θ)-10)

Name: Problem 58, Chapter 1: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-02-26T07:55:56-08:00
Duration: 1 min 23 s
Channel: Varsha Aggarwal
Description: Find the minimum value of f(θ)=(3 sin(θ)-4 cos(θ)-10) (3 sin(θ)+4 cos(θ)-10)

step 1 So f theta equal, f theta equal 9, sine square theta minus 16, cos squared theta, minus 10, 3, s

Find the minimum value of f(θ)=(3 sin(θ)-4 cos(θ)-10) (3...

Key Concepts

Trigonometric Identities

Trigonometric identities, such as the Pythagorean identity, are fundamental tools used to manipulate and simplify expressions involving sine and cosine. These identities allow one to relate different trigonometric functions to each other and to rewrite complex combinations in a simpler form, which is key in solving equations and optimizing functions in trigonometric contexts.

Amplitude-Phase Transformation

The amplitude-phase transformation is a method used to rewrite expressions of the form a sin(?) ± b cos(?) into a single trigonometric function, typically in the form R sin(? ± ?) or R cos(? ± ?), where R is the amplitude and ? is the phase shift. This conversion simplifies handling the oscillatory behavior of the function and aids in determining its extremal values.

Difference of Squares Identity

The difference of squares identity states that (A - B)(A + B) equals A² - B². This algebraic property is often used to simplify products of two binomials that are conjugates of each other, reducing the problem complexity by transforming the product into the difference between the squares of the terms.

Range of Trigonometric Functions

Understanding the range of sine and cosine functions is essential, as these functions are bounded between -1 and 1. This knowledge helps in determining the maximum and minimum values of expressions involving these functions, as scaling and shifting modify the range accordingly and are central to optimization tasks.

Optimization Techniques

Optimization techniques involve methods to determine the maximum or minimum values of a function. In the context of trigonometric functions, one often uses transformations, identities, or calculus (by finding derivatives and setting them to zero) to locate the optimal points. Recognizing the structure of the function, such as symmetry or periodic behavior, is a critical step in effectively applying these methods.

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