Hayaan
Ipakita na cos²π / 10 + cos²4π / 10 + cos² 6π / 10 + cos²9π / 10 = 2. Ako ay medyo nalilito kung gumawa ako Cos²4π / 10 = cos² (π-6π / 10) & cos²9π / 10 = cos² (π-π / 10), ito ay magiging negatibo bilang cos (180 ° -theta) = - costheta sa ang pangalawang kuwadrante. Paano ko mapapatunayan ang tanong?
Mangyaring tingnan sa ibaba. LHS = cos ^ 2 (pi / 10) + cos ^ 2 ((4pi) / 10) + cos ^ 2 ((6pi) / 10) + cos ^ 2 ((9pi) / 10) 10) + cos ^ 2 (4pi) / 10) + cos ^ 2 (pi- (4pi) / 10) + cos ^ 2 (pi- (pi) / 10) = cos ^ 2 (pi / 10) + cos ^ 2 (pi / 10) + cos ^ 2 ((4pi) / 10) = 2 * [cos ^ 2 (pi / 10) + cos ^ 2 ((4pi) 10)] = 2 * [cos ^ 2 (pi / 2- (4pi) / 10) + cos ^ 2 ((4pi) / 10)] = 2 * [sin ^ 2 ((4pi) / 10) + cos ^ 2 ((4pi) / 10)] = 2 * 1 = 2 = RHS
Kung 2sin theta + 3cos theta = 2 patunayan na 3sin theta - 2 cos theta = ± 3?
Mangyaring tingnan sa ibaba. Given rarr2sinx + 3cosx = 2 rarr2sinx = 2-3cosx rarr (2sinx) ^ 2 = (2-3cosx) ^ 2 rarr4sin ^ 2x = 4-6cosx + 9cos ^ 2x rarrcancel (4) -4cos ^ 2x = cancel (4) - 6cosx + 9cos ^ 2x rarr13cos ^ 2x-6cosx = 0 rarrcosx (13cosx-6) = 0 rarrcosx = 0,6 / 13 rarrx = 90 ° Ngayon, 3sinx-2cosx = 3sin90 ° -2cos90 ° = 3
Ipakita na, (1 + cos theta + i * sin theta) ^ n + (1 + cos theta - i * sin theta) ^ n = 2 ^ (n + 1) * (cos theta / 2) ^ n * cos n * theta / 2)?
Mangyaring tingnan sa ibaba. (1 + costheta) ^ 2 + sin ^ 2theta) = sqrt (2 + 2costheta) = sqrt (2 + 4cos ^ 2 (theta / 2 ) 2) = 2cos (theta / 2) at tanalpha = sintheta / (1 + costheta) == (2sin (theta / 2) cos (theta / 2) (theta / 2) o alpha = theta / 2 pagkatapos 1 + costheta-isintheta = r (cos (-alpha) + isin (-alpha)) = r (cosalpha-isinalpha) at maaari naming isulat (1 + costheta + ^ n + (1 + costheta-isintheta) ^ gamit ang teorem ng DE MOivre bilang r ^ n (cosnalpha + isinnalpha + cosnalpha-isinnalpha) = 2r ^ ncosnalpha = 2 * 2 ^ ncos ^ n (theta / 2) cos ((ntheta) / 2) = 2 ^ (n + 1) cos ^ n (theta / 2) cos ((ntheta) / 2)