In today's blog, I show two lemmas that are useful for converting between sin and cosine.
Lemma 1: cos(x) = sin(π/2 - x)
Proof:
(1) cos(x) = sin(x + π/2) [See Lemma 1, here]
(2) sin(x + π/2) = -sin(-x -π/2) [See Property 4, here]
(3) -sin(-x - π/2) = sin(-x - π/2 + π) [See Corollary 1.2, here]
(4) sin(-x -π/2 + π) = sin(π/2 - x)
QED
Corollary 1.1: cos(π/2 - x) = sin(x)
Proof:
(1) cos(x) = sin(π/2 - x) [See Lemma 1 above]
(2) Let x = π/2 - z
(3) cos(π/2 - z) = sin(π/2 - [π/2 - z]) = sin(z)
QED
