Testwiki:Reference desk/Archives/Mathematics/2013 November 24

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November 24

Angle sum diagram question

Trigonometric angle additive formula

I found the following image to explain the trigonometric angle additive formula. Sadly there is no explanation to go along with it. Could someone try to explain it a little better? - Letsbefiends (talk) 12:27, 24 November 2013 (UTC)

The line in bold with the 1 on it is of length 1. It is the hypotenuse of a right angle triangle with angle β which gives the sin β and cos β. The cos β line is the hypotenuse of a right angle triangle with angle α so it has sides sin α and cos α both multiplied by cos β. Same thing with the sin β line. The original line is also the hypotenuse of a right angle triangle with angle α+β, the opposite side is the sin(α+β) line up from the origin and the adjacent side is the cos(α+β) segment going horizontally from the top left.

Overall the diagram can be used to show the sine and cosine of sum identities

$\sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta$

$\cos (\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta$

because the opposite sides of the rectangle are equal. Hope that helps Dmcq (talk) 13:09, 24 November 2013 (UTC)

Oh!!! That's a great explanation! Thanks :-) - Letsbefiends (talk) 11:56, 25 November 2013 (UTC)