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Trigonometry Puzzle

Chris Smith (@aap03102) produces a weekly Maths newsletter, which is fantastic. If you don’t already subscribe get in touch with him and get on the list. One of my favourite parts of the newsletter is the puzzle of the week, and this week’s was a particular nice one:

image

Have you solved it yet? If not have a think before reading on….

I solved this within a few seconds, but still appreciate how nice a puzzle it is, and think I may only have solved it so fast because of a conversation I’d had Friday morning. On Friday morning I’d been teaching mechanics, and one question on resolving forces had an answer “Sin x”. When I revealed the answer one of my students said “I guess that would work as well.” Intrigued I ventured over and saw he had the answer “Cos (90 – x)”. This led to a nice class discussion of this property, a proof of it and the class attempting that question from THAT C3 exam. All this meant that this result was at the forefront of my mind, so when I saw the puzzle I immediately realised that

Sin^2(1) + sin^2(89) = Sin^2(1) + cos^2(1) = 1

And so on for 44 pairs, this just leaves sin^2(45), which is 1/2 as sin45=1/rt2.

This gives a nice final answer of 44.5, which Chris promptly confirmed. A lovely little puzzle.

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