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In the XVI century astronomers derived the following identities, which were called Prosthaphaere sis formulas (from the Greek Prosthesis = Addition and Apharesis Subtraction) cos(a-3) cos(a + B) (1) sin a sin 3 2 (2) (3) cos a cos sin a cos cos(a3) cos(a + 3) 2 sin(a + 3) + sin(a -3) 2 sin(a + 3) sin(a - 3) (4) cos a sin 3 = 2 You can find a purely geometric proof here. Here, I want you to prove them using algebraic means, namely exploiting Euler's formula and the algebra of complex numbers.

In the XVI century astronomers derived the following identities, which were called Prosthaphaere sis formulas (from the Greek Prosthesis = Addition and Apharesis Subtraction) cos(a-3) cos(a + B) (1) sin a sin 3 2 (2) (3) cos a cos sin a cos cos(a3) cos(a + 3) 2 sin(a + 3) + sin(a -3) 2 sin(a + 3) sin(a - 3) (4) cos a sin 3 = 2 You can find a purely geometric proof here. Here, I want you to prove them using algebraic means, namely exploiting Euler's formula and the algebra of complex numbers.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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In the XVI century astronomers derived the following identities, which were called Prosthaphaere- sis formulas (from the Greek Prosthesis = Addition and Apharesis = Subtraction) (1) sin a sin 3 cos(a) cos(a + B) 2 cos(a -3) - cos(a + B) 2 (2) (3) cos a cos 3 sin a cos 3 = = sin(a + 3) + sin(a - 3) 2 (4) cos a sin sin(a + 3) - sin(a B) 2 You can find a purely geometric proof here. Here, I want you to prove them using algebraic means, namely exploiting Euler's formula and the algebra of complex numbers. Reflect on the two proofs (geometric and algebraic) and comment on which one you find easier. Based on the formulas above, if f(t) and g(t) are two signals with a strong spectral peak at the same frequency w, what can you say about the spectrum of the signal h(t) = f(t)g(t)?
Transcribed Image Text:In the XVI century astronomers derived the following identities, which were called Prosthaphaere- sis formulas (from the Greek Prosthesis = Addition and Apharesis = Subtraction) (1) sin a sin 3 cos(a) cos(a + B) 2 cos(a -3) - cos(a + B) 2 (2) (3) cos a cos 3 sin a cos 3 = = sin(a + 3) + sin(a - 3) 2 (4) cos a sin sin(a + 3) - sin(a B) 2 You can find a purely geometric proof here. Here, I want you to prove them using algebraic means, namely exploiting Euler's formula and the algebra of complex numbers. Reflect on the two proofs (geometric and algebraic) and comment on which one you find easier. Based on the formulas above, if f(t) and g(t) are two signals with a strong spectral peak at the same frequency w, what can you say about the spectrum of the signal h(t) = f(t)g(t)?
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