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

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

$$\qquad \qquad \qquad \qquad \qquad$$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?

## closed as off-topic by Adrian Keister, Namaste, max_zorn, Sil, onurcanbektasAug 14 '18 at 7:21

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Adrian Keister, Namaste, max_zorn, Sil, onurcanbektas
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By using $$ab\leq |a||b|\leq \frac{a^2+b^2}{2}$$ we derive that:
$$f(n)=\sum_{i=1}^{n-1} \sin x_i\cos x_{i+1}+\sin x_n\cos x_1\leq \sum_{i=1}^{n-1}\frac{\sin^2x_i+\cos^2x_{i+1}}{2}+\frac{\sin^2x_n+\cos^2x_1}{2}=\sum_{i=1}^{n}\frac{\sin^2x_i+\cos^2x_i}{2}=\frac{n}{2}$$
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.