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Valentin Fadeev

Simply complex

Sunday, 15 Aug 2010, 14:53
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Edited by Valentin Fadeev, Friday, 24 Sept 2010, 13:25

Discussing one excercise on the forum recently we disagreed on the point, whether turning to complex numbers makes the solution more or less straightforward.

Here I'm digging out an example showing that this tescnique is not always as obscure as it sounds. And yes, this is another example of inappropriate use of fine methods against a basic school problem:

a times sine of t plus b times cosine of t equals c

sine of t equals e super i times t minus e super negative i times t divided by two times i

cosine of equals e super i times t plus e super negative i times t divided by two

e super i times t equals z

equation left hand side a times open z minus one divided by z close plus b times i times open z plus one divided by z close equals right hand side two times c times i

open a plus b times i close times z squared minus two times c times i times z minus open a minus b times i close equals zero

cap d equals a squared plus b squared minus c squared

equation sequence z sub one comma two equals c times i plus minus Square root of cap d divided by a plus b times i equals negative b times c plus minus a times Square root of cap d divided by a squared plus b squared plus a times c minus or plus b times Square root of cap d divided by a squared plus b squared times i

equation sequence z equals e super i times t equals cosine of t plus i times sine of t

cosine of t equals negative b times c plus minus a times Square root of cap d divided by a squared plus b squared

sine of t equals a times c minus or plus b times Square root of cap d divided by a squared plus b squared

Tags: trigonometric equations, complex numbers
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