Hello everyone I am new here. I am studying engineering and I have taken a Mathematics course as a major. I am interested in studying geometry in future. I was solving a trigonometric problem and stuck after some stages. I need to show the below inequality to complete the original problem. Please help me to solve the inequality:
$$\left|\cos\left(\frac{(1-2r)\pi}{2n}\right)\right|\leq\left|\cos\frac{\pi}{2n}\right|, \quad\mathrm{where} \quad r, n\in \mathbb{N} \quad \mathrm{and} \quad 1\leq r\leq 2n.$$
By drawing graph I can see that the inequality holds correctly but I am unable to solve using mathematical calculation. I shall be really grateful if anyone can help me to solve the inequality. Thank you in advance
The values of your LHS are: $\left|\frac{\cos(2k+1)π}{2n}\right|$ for $k=0,…,n−1.$ Use that $\cos$ is decreasing on $[0,π]$ and you will find an upper and lower bound for these $n$ $\cos$ 's.