Math Question and Answers


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 162344 by stelor last updated on 28/Dec/21

Commented by mr W last updated on 29/Dec/21

$t w o e q u a t i o n s f o r t h r e e u n k n o w n s$ $\Rightarrow n o u n i q u e s o l u t i o n!$
	Answered by aleks041103 last updated on 29/Dec/21

	$c o s a + c o s b + c o s c = 0 (1)$ $s i n a + s i n b + s i n c = 0 (2)$ $\Rightarrow e^{i a} + e^{i b} + e^{i c} = 0$ $\Rightarrow e^{i (a - c)} + e^{i (b - c)} + 1 = 0$ $\Rightarrow e^{i (a - c)} = {(e^{i (b - c)})}^{*} = e^{i (c - b)}$ $a - c = c - b$ $a + b = 2 c$ $a l s o$ $- \frac{1}{2} = c o s (a - c)$ $\Rightarrow a - c = \frac{2 π}{3} a n d \frac{4 π}{3}$ $\Rightarrow a = c \pm \frac{2 π}{3}$ $\Rightarrow b = c \pm (- \frac{2 π}{3})$
	Commented by aleks041103 last updated on 29/Dec/21

	$t h i s h a s a c u t e g e o m e t r i c p r o o f t o o .$ $t h e c o m p l e x v e c t o r s e^{i a}, e^{i b}, e^{i c} m u s t$ $s u m u p t o 0 a n d h a v e t h e s a m e l e n g t h .$ $T h e r e f o r e t h e t r e e v e c t o r s m u s t b e s y m m e t r i c$ $\Rightarrow t h e a n g l e b e t w e e n a n y t w o o f t h e m$ $m u s t b e \frac{360 °}{3} = 120 ° .$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum