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AoMath: Trig Identities



Note: This does NOT prove the trig identities but it Does help us memorize them.



a = cos(G), b = sin(G), c = cos(H), d = sin(H)

(a + bi)(c + di) has angle G + H
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
(ac - bd) + (ad + bc)i = (cos(G)cos(H) - sin(G)sin(H)) + (cos(G)sin(H) + sin(G)cos(H))i
(ac - bd) + (ad + bc)i = cos(G + H) + isin(G + H)
(cos(G)cos(H) - sin(G)sin(H)) + (cos(G)sin(H) + sin(G)cos(H))i = cos(G + H) + isin(G + H)

cos(G)cos(H) - sin(G)sin(H) = cos(G + H)
cos(G)sin(H) + sin(G)cos(H) = sin(G + H)