Let n be a fixed positive integer such that sin((π/(2n)))+cos((𝛑/(2n)))=((√n)/2) Then find n


Question and Answers Forum

All Questions Topic List
None Questions
Previous in All Question Next in All Question
Previous in None Next in None
Question Number 193494 by aba last updated on 15/Jun/23

$Let n be a fixed positive integer such that$ $\sin (\frac{π}{2 n}) + \cos (\frac{π}{2 n}) = \frac{\sqrt{n}}{2}$ $Then find n$
Commented by maths_plus last updated on 15/Jun/23

$cool$
	Answered by Frix last updated on 15/Jun/23

	$n = 6$
	Answered by witcher3 last updated on 15/Jun/23
	$∣cos(a)+sin(a)∣=(√(2⟨∣))sin(a+(π/4))∣≤(√2) ⇒((√n)/2)≤(√2)⇒n≤8 n∈{1,....8} (sin((π/(2n)))+cos((π/(2n))))^2 =1+sin((π/n))=(n/4) sin((π/n))=((n−4)/4)..E sin(x)≤x⇒((n−4)/4)≤(π/n)≤(4/n)⇒n^2 −4n−16≤0 n∈{((4−(√(80)))/2),((4+(√(80)))/2)}⇒n≤((13)/2)⇒n≤6 sin n∈{1....8}⇒sin((π/n))>0..E⇒n−4>0⇒n≥5 n∈{5,6} simply check n=6 unique solution$
	$∣ \cos (a) + \sin (a) ∣= \sqrt{2 ⟨ ∣} \sin (a + \frac{π}{4}) ∣⩽ \sqrt{2}$ $\Rightarrow \frac{\sqrt{n}}{2} ⩽ \sqrt{2} \Rightarrow n ⩽ 8$ $n \in {1, . . . . 8}$ ${(\sin (\frac{π}{2 n}) + \cos (\frac{π}{2 n}))}^{2} = 1 + \sin (\frac{π}{n}) = \frac{n}{4}$ $\sin (\frac{π}{n}) = \frac{n - 4}{4} . . E$ $\sin (x) ⩽ x \Rightarrow \frac{n - 4}{4} ⩽ \frac{π}{n} ⩽ \frac{4}{n} \Rightarrow n^{2} - 4 n - 16 ⩽ 0$ $n \in {\frac{4 - \sqrt{80}}{2}, \frac{4 + \sqrt{80}}{2}} \Rightarrow n ⩽ \frac{13}{2} \Rightarrow n ⩽ 6$ $\sin n \in {1 . . . . 8} \Rightarrow \sin (\frac{π}{n}) > 0 . . E \Rightarrow n - 4 > 0 \Rightarrow n ⩾ 5$ $n \in {5, 6}$ $simply check n = 6 unique solution$
	Commented by York12 last updated on 16/Jun/23

	$W i t c h e r 3 a r e y o u u s i n g d i s c o r d$
	Answered by MM42 last updated on 15/Jun/23

	$1 + s i n (\frac{π}{n}) = \frac{n}{4} \Rightarrow s i n (\frac{π}{n}) = \frac{n - 4}{4}$ $0 ⩽ \frac{n}{4} - 1 ⩽ 1 \Rightarrow 4 ⩽ n ⩽ 8$ $o n l y n = 6 t h e r e i s a t i e$
	Commented by aba last updated on 15/Jun/23
	��
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum