A' cleachdadh foirmlean ceàrn dùbailte
Tha e math fios a bhith agad, nuair a tha thu a' fuasgladh co-aontar triantanachd sam bith anns a bheil \(\sin 2x\) agus an dara cuid \(\sin x\) no \(\cos x\) gur e an aon phròiseas a th' ann. An àite \(\sin 2x\) cuir ann \(2\sin x\cos x\), thoir na teirmean gu lèir gu aon taobh, factaraichCuir abairt ann an camagan. Mar eisimpleir, 18x + 12y = 6(3x + 2y). 'S e factaradh am pròiseas meudachaidh a' dol an rathad eile. agus fuasgail.
Question
Fuasgail an co-aontar \(5\sin 2x^\circ + 7\cos x^\circ = 0\) airson \(0^\circ \le x^\circ \le 360^\circ\)
\(5\sin 2x^\circ + 7\cos x^\circ = 0\)
An àite \(\sin 2x\) cuir ann \(2\sin x\cos x\)
\(5(2\sin x^\circ \cos x^\circ ) + 7\cos x^\circ = 0\)
\(10\sin x^\circ \cos x^\circ + 7\cos x = 0\)
Thoir a-mach \(\cos x^\circ\) mar factar cumantaÀireamh shlàn as urrainn dhuinn a roinn gu mionaideach ann an dà (no barrachd) àireimh eile, me tha 4 na fhactar cumanta de 8, 12 agus 20., ach na roinn leis an fhactar chumanta.
\(\cos x^\circ (10\sin x^\circ + 7) = 0\)
Mar sin \(\cos x^\circ = 0\) no \(10\sin x^\circ + 7 = 0\)
Fuasgail gach co-aontar, fear mu seach:
\(\cos x^\circ = 0\)
\(x^\circ = 90^\circ \,or\,270^\circ\)
agus:
\(10\sin x^\circ + 7 = 0\)
\(\sin x^\circ = - \frac{7}{{10}}\)
\(x^\circ = 224.4^\circ \,or\,315.6^\circ\)
Tha sin a' toirt dhuinn nam fuasglaidhean \(90^\circ ,224.4^\circ ,270^\circ ,315.6^\circ\)
Tha dà dhòigh eile a dh'fheumas tu ionnsachadh agus tha na dhà co-cheangailte ri \(\cos 2x\):
- ann an co-aontar triantanachd sam bith anns a bheil \(\cos 2x\) agus \(\sin x\), an àite \(\cos 2x\) cuir ann \(1 - 2{\sin ^2}x\)
- ann an co-aontar triantanachd sam bith anns a bheil \(\cos 2x\) agus \(\cos x\), an àite \(\cos 2x\) cuir ann \(2{\cos ^2}x - 1\)
An uair sin, anns an dà shuidheachadh, thoir na teirmean gu lèir gu aon taobh, a' cruthachadh co-aontar ceàrnanach ann an teirmean \(\sin x\) no \(\cos x\) a-mhain agus an uair sin fuasgail.
Question
Fuasgail \(\cos 2x + 3\sin x + 1 = 0\) airson \(0\textless x\textless2\pi\)
\(\cos 2x + 3\sin x + 1 = 0\)
\(1 - 2{\sin ^2}x + 3\sin x + 1 = 0\)
\(- 2{\sin ^2}x + 3\sin x + 2 = 0\)
\(2{\sin ^2}x - 3\sin x - 2 = 0\)
'S e co-aontar ceàrnanach a tha seo a ghabhas fhactaradh.
\((2\sin x + 1)(\sin x - 2) = 0\)
An dara cuid:
\(2\sin x + 1 = 0\)
No:
\(\sin x - 2 = 0\)
Mar sin:
\(\sin x = - \frac{1}{2}\)
No:
\(\sin x = 2\)
Bhon a tha \(- 1 \le \sin x \le 1\), chan eil fuasgladh aig a' cho-aontar seo, agus tha sin a' toirt dhuinn:
\(x = \frac{{7\pi }}{6}\) no \(\frac{{11\pi }}{6}\)
'S e luachan mionaideach a tha seo – chan fheum thu àireamhair airson seo. Ach tha e ceart gu leòr do fhreagairt a thoirt mar dheicheadan gu 3 ionadan deicheach, nas lugha na thèid luachan mionaideach iarraidh ort.
