The benefit of this post is for me to recall better the two following formulas:
In the past, whenever I have had to find the cosine or sine of three times and angle (e.g. for some maths contest problems), I have manually performed expansions based on the double angle formulas:
Now that I see these formulas together, I want a way of remembering them without having to write this many lines. Here is my aid.
- Both formulas are of the form , where for the cosine formula and for the sine formula (I already knew this).
- The coefficients up to sign are 4 and 3 (I already knew this).
- The first coefficient is positive, the second negative (hence 4, -3 or 3, -4).
- The cubic term is paired up with 4. Alternatively, the cubic term is not paired up with 3
This last point comes about because the 2 from is added to the 2 from .
If one is unsure whether is or , simply substitute – the expression should evaluate to 1. The coefficients for are then flipped in sign.