Difreálaigh w.r.t. α
-\sin(\alpha )
Luacháil
\cos(\alpha )
Tráth na gCeist
Trigonometry
5 fadhbanna cosúil le:
\cos \alpha = ?
Roinn
Cóipeáladh go dtí an ghearrthaisce
\frac{\mathrm{d}}{\mathrm{d}\alpha }(\cos(\alpha ))=\left(\lim_{h\to 0}\frac{\cos(\alpha +h)-\cos(\alpha )}{h}\right)
Do fheidhm f\left(x\right), is ionann an díorthach agus teorainn \frac{f\left(x+h\right)-f\left(x\right)}{h} toisc go dtéann h go 0, más ann don teorainn sin.
\lim_{h\to 0}\frac{\cos(h+\alpha )-\cos(\alpha )}{h}
Úsáid an Fhoirmle Shuime don Chomhshíneas.
\lim_{h\to 0}\frac{\cos(\alpha )\left(\cos(h)-1\right)-\sin(\alpha )\sin(h)}{h}
Fág \cos(\alpha ) as an áireamh.
\left(\lim_{h\to 0}\cos(\alpha )\right)\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)-\left(\lim_{h\to 0}\sin(\alpha )\right)\left(\lim_{h\to 0}\frac{\sin(h)}{h}\right)
Athscríobh an teorainn.
\cos(\alpha )\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)-\sin(\alpha )\left(\lim_{h\to 0}\frac{\sin(h)}{h}\right)
Bain leas as an rud é go bhfuil \alpha ina thairiseach agus teorainneacha á ríomh agus h ag dul go 0.
\cos(\alpha )\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)-\sin(\alpha )
Is ionann teorainn \lim_{\alpha \to 0}\frac{\sin(\alpha )}{\alpha } agus 1.
\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)=\left(\lim_{h\to 0}\frac{\left(\cos(h)-1\right)\left(\cos(h)+1\right)}{h\left(\cos(h)+1\right)}\right)
Chun an teorainn \lim_{h\to 0}\frac{\cos(h)-1}{h} a luacháil, méadaigh an t-uimhreoir agus an t-ainmneoir faoi \cos(h)+1 ar dtús.
\lim_{h\to 0}\frac{\left(\cos(h)\right)^{2}-1}{h\left(\cos(h)+1\right)}
Méadaigh \cos(h)+1 faoi \cos(h)-1.
\lim_{h\to 0}-\frac{\left(\sin(h)\right)^{2}}{h\left(\cos(h)+1\right)}
Baint Úsáid as Aitheantas Píotagarásach.
\left(\lim_{h\to 0}-\frac{\sin(h)}{h}\right)\left(\lim_{h\to 0}\frac{\sin(h)}{\cos(h)+1}\right)
Athscríobh an teorainn.
-\left(\lim_{h\to 0}\frac{\sin(h)}{\cos(h)+1}\right)
Is ionann teorainn \lim_{\alpha \to 0}\frac{\sin(\alpha )}{\alpha } agus 1.
\left(\lim_{h\to 0}\frac{\sin(h)}{\cos(h)+1}\right)=0
Bain leas as an rud go bhfuil \frac{\sin(h)}{\cos(h)+1} leanúnach ag 0.
-\sin(\alpha )
Ionadaigh an luach 0 isteach sa slonn \cos(\alpha )\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)-\sin(\alpha ).
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Teorainneacha
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