A Curiously Little-Used Trigonometric Formula

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, Universityof Wisconsin - Green Bay

Despite the fact that this formula is simple to prove and simple in form, it rarely shows up in math references. It's a simple way to calculate the opposing angles of a triangle, given two sides and the included angle. I suspect it's because it's not as pretty and symmetrical as the cosine laws, it involves the ugly stepchild, the tangent, and it doesn't have a simple denominator.

Given a, b and C, find A.

1. h = a sin C
2. h = d tan A
3. d = b - a cos C
4. Thus a sin C = (b - a cos C) tan A  and tan A = (a sin C)/(b - a cos C)
5. Likewise tan B = (b sin C)/(a - b cos C)

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Created 25 January 2011, Last Update 15 January 2020