Small-angle approximation: Difference between revisions

Revision as of 11:10, 22 December 2023

Revision as of 03:28, 9 March 2024

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==== Dual numbers ====

ByOne usingmay theuse Maclaurin''[[dual series of cosine and sine and substituting in θ=θεnumbers]]'', wheredenoted ε is the symbol used in dual numbers, often considered similar to an infinitesimal amount, with a square of 0. By using the Maclaurin series of cosine and sine and substituting θ=θε, the result is that cos(θε)=1 and sin(θε)=θε. These approximations satisfy the [[Pythagorean Identity, as cos²(θε)+sin²(θε)=1²+(θε)²=1+θ²ε²=1+θ²0=1.identity]]:

: cos²(θε)+sin²(θε)=1²+(θε)²=1+θ²ε²=1+θ²0=1.

==Error of the approximations==