step 1 In this problem, the given equation can be rewritten as 2 sine of x is equals to 3 into 1 plus c

2 sinx-3 cosx=3 | Numerade

Name: Problem 88, Chapter 12: Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry
Uploaded: 2021-08-23T11:34:10-08:00
Duration: 1 min 32 s
Channel: Aman Gupta
Description: 2 sinx-3 cosx=3

Key Concepts

Linear Combination of Sine and Cosine

This concept deals with expressions where sine and cosine functions appear in a linear combination, typically written as A sin x + B cos x. Such expressions are common in trigonometry, and understanding them involves identifying how the coefficients A and B influence the behavior, amplitude, and phase of the combined wave.

Amplitude-Phase Transformation

This method involves rewriting a linear combination of sine and cosine into the form R sin(x + ?), where R is the amplitude, calculated as the square root of the sum of the squares of the original coefficients, and ? (the phase shift) is determined by the ratio of the coefficients. This transformation simplifies both the analysis and solution of trigonometric equations by reducing them to a single sinusoidal function.

Range of Trigonometric Functions

Understanding the range of trigonometric expressions is crucial when solving equations involving sine and cosine. For an expression rewritten as R sin(x + ?), the maximum and minimum values are R and -R respectively. This concept is used to determine whether an equation has a solution by checking if the constant on the other side of the equation lies within the interval [-R, R].

Solving Trigonometric Equations

This broad concept involves finding all angles that satisfy a given trigonometric equation. It typically includes methods such as transformation techniques (e.g., amplitude-phase transformation), utilization of trigonometric identities, and consideration of the periodicity and symmetry of sine and cosine functions. These techniques ensure a comprehensive approach to identifying all possible solutions.

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