How astronomers multiplied without multiplying | Product to sum and sum to product formulas

Here's a strange way to multiply 0.9 by 0.7: look up two numbers in a cosine table, add them, halve the answer. You get 0.63, exactly the product. In 1580, at Tycho Brahe's observatory, astronomers multiplied this way, decades before logarithms. In this video: the product-to-sum formula 2 cos x cos y = cos(x + y) + cos(x − y), built from two turns (NCERT class 11, identity 21); 0.9 × 0.7 and 0.6 × 0.8 worked the 1580 way; who really found it (Werner, Wittich, Tycho, Ursus); sum to product (identity 20), with cos 75° + cos 15° = √6/2 exactly and sin 5x + sin 3x = 2 sin 4x cos x; why two notes at 440 and 442 Hz beat twice a second, and two tuning forks at 256 and 260 Hz four times; and board-style questions: 2 sin 3x cos x, 2 cos 4x cos 2x, and the tan 2x and tan 4x proofs. Educational. Play along in the free notebook linked below: run each cell with Shift + Enter, then multiply with a cosine table and hear the beats. Check yourself: two notes at 300 Hz and 303 Hz. How many beats a second? Play along in your browser (free, nothing to install): https://nammaschool.github.io/interac... Or open it in Google Colab: https://colab.research.google.com/git... Chapters 0:00 A strange way to multiply 0:58 Two turns, added 2:08 The astronomers' machine 3:31 Who got there first 4:08 Run it backwards 6:09 A formula you can hear 7:39 What the examiner asks 9:08 Try it in the notebook 9:56 Adding, again #trigonometry #trigidentities #maths #cbse #class11 #mathhistory #nammaschool