Alternative way to remember your Trigonometric Addition formula

An illustration of a complex number plotted on...
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Not the type to remember your formula? Me too

I never remember my formula until one or two days before test and it practically vanished from my mind 5 minutes after the test is over. However, I usually have ways so I can derive them when I need them without peeking at my books.

One of them is trigonometric addition formulas.. You know cos (a+b) = … and sin (a+b) = …

My way to remember them require a little geometric intuition about complex numbers. For every complex number with modulus 1, the number can be written as cos a + i sin a (cis a) for a particular angle a where a is the angle between the vector a and positive x axis. If you multiply the number with another complex number of modulus 1, say cis b, you know that geometrically you just rotate a by angle b. That is, cis a * cis b = cis (a+b).

That way, by calculating the terms on the left side, and by grouping the terms with i and without i, you get your formula for cos (a+b) and sin (a+b) right away. Apply division, then you also get your tan (a+b) as well.