Wikipedia:Reference desk/Archives/Mathematics/2017 January 5

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January 5

Is there any reason behind this trigonometry mnemonic?

The sine of {0, pi/6, pi/4, pi/3, pi/2} is {root 0 over 2, root 1 over 2, root 2 over 2, root 3 over 2, root 4 over 2}. Thank you. 69.22.242.15 (talk) 20:24, 5 January 2017 (UTC)

Because of the Pythagorean theorem? --Jayron32 20:35, 5 January 2017 (UTC)

Substitute that into the formula cos 2A = 1 - 2 sin²A and you get:

cos {0, π / 3, π / 2, 2π / 3, π} is {2/2, 1/2, 0/2, -1/2, -2/2}

does that look more familiar or obvious?--JohnBlackburne^words_deeds 20:39, 5 January 2017 (UTC)

It does. It looks like the interaction between exponents of 2 and 1. Does it have anything to do with Spiral_of_Theodorus?69.22.242.15 (talk) 21:08, 5 January 2017 (UTC)

I think the OP was wondering if there is any intuition that would lead us to predict such a pattern before using the Pythagoren theorem on each one individually. Loraof (talk) 20:54, 5 January 2017 (UTC)

The answer I give is sort of intuitive; you can draw it as a circle, radius 1, with lines at 0, ±1, ± 1/2. Measure the angles from the top to get {0, π / 3, π / 2, 2π / 3, π}. By trigonometry the cosines of these are the heights of the lines. Finally plug those into

sin A = √1/2 (1 - cos 2A)

to get the numbers in the original question.--JohnBlackburne^words_deeds 21:11, 5 January 2017 (UTC)