Foirmlean cur-ris

Nuair a bhios sinn a' cur-ris no a' toirt-air-falbh cheàrnan canaidh sinn ceàrn dà-fhillte ris an toradh. Mar eisimpleir, 's e ceàrn dà-fhillte a th' ann an \(30^\circ + 120^\circ\). Le àireamhair, obraichidh sinn a-mach:

\[\sin (30^\circ + 120^\circ ) = \sin (210^\circ ) = - 0.5\]

\[\sin (30^\circ ) + \sin (120^\circ ) = 1.366\,(gu\,3\,id.)\]

Tha seo a' sealltainn nach eil \(\sin (A + B)\) co-ionann ri \(\sin A + \sin B\). An àite sin, faodaidh sinn na co-ionannachdan a leanas a chleachdadh:

\[\sin (A + B) = \sin A\cos B + \cos A\sin B\]

\[\sin (A - B) = \sin A\cos B - \cos A\sin B\]

\[\cos (A + B) = \cos A\cos B - \sin A\sin B\]

\[\cos (A - B) = \cos A\cos B + \sin A\sin B\]

curriculum-key-fact
  • Bidh foirmlean airson cur-ris air an sgrìobhadh ann an riochd goirid:
  • \[\sin (A \pm B) = \sin A\cos B \pm \cos A\sin B\]
  • \[\cos (A \pm B) = \cos A\cos B \pm \sin A\sin B\]

Bidh sinn a' cleachdadh nam foirmlean seo gus fuincseanan triantanachd a mheudachadh gus ar cuideachadh a' sìmpleachadh, no ag obrachadh a-mach, abairtean triantanachd dhen riochd seo.

Seo mar a bhios sinn a' dol an sàs sa cheist seo sa bheil dà phàirt:

Question

1. Sgrìobh \(75^\circ = 45^\circ + 30^\circ\) agus obraich a-mach an luach mionaideach aig \(\sin 75^\circ\)

2. Obraich a-mach an luach mionaideach aig \(\cos \left( {\frac{{7\pi }}{{12}}} \right)\)

1. \(\sin 75^\circ = \sin (45 + 30)^\circ\)

A' cleachdadh an fhoirmle \(\sin (A + B)\)

\[= \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ\]

A' cleachdadh luachan mionaideach air am bu chòir fios a bhith agad:

\[= \frac{1}{{\sqrt 2 }} \times \frac{{\sqrt 3 }}{2} + \frac{1}{{\sqrt 2 }} \times \frac{1}{2}\]

\[= \frac{{\sqrt 3 }}{{2\sqrt 2 }} + \frac{1}{{2\sqrt 2 }}\]

\[= \frac{{\sqrt 3 + 1}}{{2\sqrt 2 }}\]

2. Bhon a tha \(\frac{{7\pi }}{{12}} = \frac{\pi }{3} + \frac{\pi }{4}\) tha:

\[\cos \left( {\frac{{7\pi }}{{12}}} \right) = \cos \left( {\frac{\pi }{3} + \frac{\pi }{4}} \right)\]

A' cleachdadh an fhoirmle airson \(\cos (A + B)\)

\[= \cos \frac{\pi }{3}\cos \frac{\pi }{4} - \sin \frac{\pi }{3}\sin \frac{\pi }{4}\]

A' cleachdadh luachan mionaideach air am bu chòir fios a bhith agad:

\[= \frac{1}{2} \times \frac{1}{{\sqrt 2 }} - \frac{{\sqrt 3 }}{2} \times \frac{1}{{\sqrt 2 }}\]

\[= \frac{{1 - \sqrt 3 }}{{2\sqrt 2 }}\]