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The Complete Idiot's Guide to Trigonometry by Izolda Fotiyeva, Dmitriy Fotiyev

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Double-Angle Formulas

Let’s first deal with double-angle formulas, which break an expression involving a single angle into one involving two angles, and prove at least one of them. The proof is short and easy, which is a good starting point for this chapter.

Prove that sin2u = 2sinucosu.

To prove this identity, let v = u in the sum formulas for sines. Then obtain the following:

sin2u = sin(u + u) = sinucosu + cosusinu = 2sinucosu

Note that the order of the factors does not change the product; thus sinucosu = cosusinu.

You proved your first double-angle formula. The other double-angle formulas can be proved in a similar way. Let’s list all three of them.

The following are called double-angle formulas:

sin2u = 2sinucosu

cos2u = cos2u – sin2u = ...

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