$f(n) = \sin(x_1)\cos(x_2) + \sin(x_2)\cos(x_3) +\cdots+ \sin(x_n)\cos(x_1)$, where $x_1 ,\ldots, x_n$ are any real numbers

Question

$$f(n) = \sin(x_1)\cos(x_2) + \sin(x_2)\cos(x_3) +\cdots+ \sin(x_n)\cos(x_1)$$

$\qquad \qquad \qquad \qquad \qquad $where $x_1 ,\ldots, x_n$ are any real numbers.

State which one(s) is/are correct:

(A) $f(100) < 100$

(B) $f(50) < 100$

(C) $f(100) > 100$

(D) $f(100) > 50$

My question is how do you solve or analyze these type of problems?

Answer 1

By using $ab\leq |a||b|\leq \frac{a^2+b^2}{2}$ we derive that:

$f(n)=\sum_{i=1}^{n-1} \sin x_i\cos x_{i+1}+\sin x_n\cos x_1\leq \sum_{i=1}^{n-1}\frac{\sin^2x_i+\cos^2x_{i+1}}{2}+\frac{\sin^2x_n+\cos^2x_1}{2}=\sum_{i=1}^{n}\frac{\sin^2x_i+\cos^2x_i}{2}=\frac{n}{2}$

Thus, (A,B) are correct, and (C,D) are not.

Answer 2

Sine and cosine both have maximum values of $1$. Multiplying them does not change this. So $f(100)$ is the addition of less than 100 terms that cannot individually be greater than $1$. Did you notice this? Apply similar logic to choice B and notice that these are mutually exclusive to C and D respectively.