The basic rules are:Originally Posted by thebearot
We get the "double angle" formulae from these:
Hint: It takes a bit of work, but you can simplify sin(3t) as sin(2t+t) and cos(4t) as cos(2t+2t).
-Dan
Hello, ThePerfectHacker!
There is an elegant way to derive a general formula for:
I "discovered" a method while in college . . . with my very own ritual.
Suppose we want the trig formulas for
(1) Expand
(2) Construct a fraction with these terms,
Write the terms alternately in the denominator and numerator.
Start with the denominator.
. .
(3) Insert alternating signs in the numerator and the denominator.
. .
(4) Replace with and we have:
. .
(5) Multiply top and bottom by and we get two formulas:
. .
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
This is all possible because of DeMoivre's Theorem.
Since , we can expand the right side,
then equate real and imaginary components, giving us (2) and (3) above.