Any periodic function can be expressed as a.a)superposition of sine and cosine functions of different time periods with suitable coefficientsb)superposition of exponential functions of real time periods with suitable coefficientsc)superposition of polynomial functions of different time periods with suitable coefficientsd)superposition of sine and cosine functions of same time period with suitable coefficientsCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

Explanation:

A periodic function is a function that repeats its values in regular intervals. It can be expressed as a superposition of sine and cosine functions of different time periods with suitable coefficients. Let's break down each part of the statement to understand why option 'A' is the correct answer.

Periodic Function:

A periodic function is a function that repeats its values after a certain time period. This means that for any value of x, the function value at x + T is the same as the value at x, where T is the time period.

Sine and Cosine Functions:

Sine and cosine functions are trigonometric functions that oscillate between -1 and 1 as the input varies. They have a periodicity of 2π, which means they repeat their values every 2π units. The sine function has an initial value of 0 at x = 0, while the cosine function has an initial value of 1 at x = 0.

Superposition:

The superposition principle states that the sum of any two or more functions can be expressed as a single function. In the case of periodic functions, we can express them as a superposition of sine and cosine functions.

Representation of Periodic Function:

Any periodic function f(x) can be represented as:

f(x) = a₀/2 + ∑(aₙcos(nx) + bₙsin(nx))

where a₀, aₙ, and bₙ are coefficients, and n is a positive integer representing the harmonic number.

Explanation of Option 'A':

Option 'A' states that any periodic function can be expressed as a superposition of sine and cosine functions of different time periods with suitable coefficients. This aligns with the representation of periodic functions mentioned above, where the function is expressed as a sum of cosine and sine terms.

By adjusting the coefficients aₙ and bₙ, we can determine the amplitudes and phases of each harmonic in the function. This allows us to match the periodic function's shape and characteristics by choosing the appropriate coefficients for each harmonic.

Therefore, option 'A' is the correct answer as it accurately describes the representation of a periodic function as a superposition of sine and cosine functions with suitable coefficients.

Answer

All periodic functions can, in principle, be represented as a sum of sine and cosine functions, since they form what's known as a complete set.

Similar Class 11 Doubts

Coefficient of static friction between shoes of an athlete and ground is 0.6 whereas coefficient of kinetic friction is 0.5, taking g =10m/s2; what can be maximum horizontal component of acceleration of the athlete in a sprint?

In the expansion of ( 1 +

The coefficients of 3 consecutive terms in the expression of (1

In the expansion of (1 x)^n the binomial coefficients of the three consecutive terms are 220,495,792 find the value of n?

Describe any four properties of the Correlation Coefficient .?

Top Courses for Class 11View all

Chemistry Class 11

CUET Mock Test Series

English Class 11

Chemistry Practice Tests: CUET Preparation

CUET Mock Test: Humanities Subjects 2025

Top Courses for Class 11

Top Courses for Class 11

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams