The correct answer is: A. P and S
A real-valued periodic function $f(x)$ can be represented as a Fourier series, which is a sum of sine and cosine functions with frequencies that are integer multiples of the fundamental frequency. The Fourier series of a real-valued even function $f(x)$ has only cosine terms, while the Fourier series of a real-valued odd function $f(x)$ has only sine terms.
Here is a more detailed explanation of each option:
- P. Cosine terms if it is even
A real-valued even function $f(x)$ is a function that satisfies $f(-x) = f(x)$. For example, the function $f(x) = x^2$ is an even function. The Fourier series of an even function has only cosine terms. This is because the cosine function is an even function, and the Fourier series is a sum of sine and cosine functions.
- Q. Sine terms if it is even
A real-valued odd function $f(x)$ is a function that satisfies $f(-x) = -f(x)$. For example, the function $f(x) = x$ is an odd function. The Fourier series of an odd function has only sine terms. This is because the sine function is an odd function, and the Fourier series is a sum of sine and cosine functions.
- R. Cosine terms if it is odd
This statement is incorrect. A real-valued odd function $f(x)$ has only sine terms, not cosine terms.
- S. Sine terms if it is odd
This statement is correct. A real-valued odd function $f(x)$ has only sine terms.