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derivatives of sine and cosine
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(Derivation)
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The derivation of the derivatives of sine and cosine is a bit simpler by using the prosthaphaeresis formulas
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Let $x,\,t$ be any real numbers such that $t \neq x$ . Then we obtain $$\frac{\sin{x}-\sin{t}}{x-t} = \frac{2\sin \left( \frac{x-t}{2} \right) \cos \left( \frac{x+t}{2} \right) }{x-t} = \frac{\sin \left( \frac{x-t}{2} \right) }{\left( \frac{x-t}{2} \right) }\cdot\cos \left( \frac{x\!+\!t}{2} \right) \;\; \longrightarrow\; 1\cdot\cos \left( \frac{x\!+\!x}{2} \right) = \cos{x},$$ as $t\to x$ . Here we used the known limit $\displaystyle\lim_{u\to0}\frac{\sin{u}}{u} = 1$ (see this entry).
The derivative of cosine is calculated similarly: $$\frac{\cos{x}-\cos{t}}{x-t} = \frac{-2\sin \left( \frac{x+t}{2} \right) \sin\left( \frac{x-t}{2} \right)}{x-t} =-1 \cdot \frac{\sin\left( \frac{x-t}{2} \right) }{\left( \frac{x-t}{2} \right) }\cdot \sin \left( \frac{x\!+\!t}{2} \right) \;\; \longrightarrow\; -1 \cdot 1\cdot \sin \left( \frac{x\!+\!x}{2} \right) =-\sin{x},$$ as $t\to x$ .
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"derivatives of sine and cosine" is owned by Wkbj79. [ full author list (2) | owner history (1) ]
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Cross-references: cosine, derivative, limit, real numbers, Prosthaphaeresis formulas
There are 2 references to this entry.
This is version 7 of derivatives of sine and cosine, born on 2007-04-24, modified 2007-04-26.
Object id is 9259, canonical name is DerivativesOfSineAndCosine.
Accessed 4479 times total.
Classification:
| AMS MSC: | 26A09 (Real functions :: Functions of one variable :: Elementary functions) |
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Pending Errata and Addenda
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