Bob
One final thought: The compound angle formulas work whatever the angles used. They aren't limited to acute angles. The proof above only works for acute angles; in fact, the diagram breaks down if A + B is over 90. I think that is why I prefer other proofs that are more general. The rotational matrix proof is my favourite.
]]>If you put alpha = beta = A, you get the double angle formulas.
eg.
Now put theta = 2A and you can re-arrange to get
and so on.
Is that what you wanted?
Bob
]]>Well done!
Bob
]]>Bob
]]>I did wonder if that's what you wanted. I could remember the formulas I gave easily so I took the line of least effort.
There are many ways of proving the compound angle formulas. The trig. approach is the first I met whilst still at school, but I haven't used it since. So I've had to dig deep in my memory.
So, make a triangle ABC with angle A = alpha and angle B = 90
Make a perpendicular line CD so that angle DCA = 90.
Choose the position of D so that DAC = beta.
Draw DF perpendicular to AB with F on AB, and finally CE perpendicular to DF with E on DF.
Call the point where DF and AC cross point G.
AGF = CGD = 90 minus alpha so GDC = alpha.
note EF = CB
The cosine formula is very similar. I'll leave it as an exercise.
hint AF = AB - FB
Bob
]]>I already did in the other thread. It looks like it is a tough nut to crack.
]]>I'll post a note to MIF.
Bob
]]>I've edited them to dotplus for now.
Bob
]]>Why is "plus" written everywhere "+" should go.
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