Let x1, x2, …∈(0, π) denote the of values of x satisfying the equation then the value of (1)/(π)

Let x1, x2, …∈(0, π) denote the of values of x satisfying...

Key Concepts

Symmetry and Periodicity in Trigonometric Functions

The inherent symmetry and periodicity of trigonometric functions often lead to patterns in their roots. By recognizing these patterns, one can simplify the process of summing these values or understanding their distribution over an interval such as (0, ?). This concept is crucial for efficiently handling and simplifying problems involving the infinite roots of trigonometric equations.

Infinite Sequences and Series

When the solutions of an equation form an infinite sequence, understanding the properties of infinite series becomes essential. Summing these roots, especially when they are transformed (e.g., divided by ?), may involve analyzing convergence and symmetry properties to obtain a finite, meaningful result.

Roots of Transcendental Equations

Many equations in mathematics are transcendental, meaning they cannot be solved using algebraic methods alone. Instead, techniques such as graphical analysis, numerical approximation, or fixed-point iteration are used to understand and approximate the roots. In this context, finding all values of x that satisfy a transcendental trigonometric equation means carefully examining how the function behaves within the interval.

Trigonometric Equations

Trigonometric equations involve functions such as sine, cosine, and tangent, which are periodic and often yield infinitely many solutions. Understanding the behavior of these functions on specific intervals, like (0, ?), is crucial for identifying all valid roots and leveraging their symmetry and periodicity properties.

Transcript

00:01 Hi, it is given that x1 x2 belongs to 0 to pi, not including 0 and pi not the values of x that is fine the equation was the equation is given here you need to find out one over pi x1 plus x2 and so on so from this equation you need to find out x1 and x2 and then evaluate this expression let's first find out this series here so let's work on this is given as 1 plus modulus cos x plus plus co square x plus modulus cos x cube and so on we have till infinity so it can be given us one plus modulus cos x plus now co square x is always positive even if put modulus cos x here and do it square then nothing will change because result here is positive here also we here, cosx is coming out to be modulus process is positive and it's square it's coming out to be positive.

01:07 So we can write like this plus we have sound till infinity.

01:13 Now it is a gp here, infinite gp and some of the series is given as a over 1 negative r.

01:21 The first term here we have 1.

01:24 R is modulus cosx.

01:28 Let's look at here if r is less than 1.

01:30 Here we have x belong to 0 to pi.

01:34 So 0 to pi we have cos x.

01:37 It's always less than 1.

01:39 So we get r less than 1.

01:44 So we can apply s infinity.

01:47 That is given as 1 over 1 negative modulus cos x.

01:54 Let's put back that in this equation.

02:00 So we get 27 is 3 cube.

02:03 So we get 3 to the power 3 over 1 negative modulus cosx equals 9 cube is given as 3 square cube that will be 3 to the power 6 compare this so the exponents are equal so we get 3 over 1 negative modulus 4 6 equals 6 so from here we get more cos x equal 1 over 2.

02:40 Now more corset, more cosex is 1 over 2.

02:44 That means cosex can take positive value or negative value.

02:48 So we have cosex is given as 1 by 2 and also negative 1 over 2.

02:59 Now if you just look at the graph 4 .0 2 pi by 2 for cosex the graph is like this...

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