Question

How do I get from 2sin(x).cos(x) to sin(2x) and please don't tell me: "it is a fundamental trig indentity". I want to be shown the derivation, with some penetrating commentary, please, with sugar on top.

Answer 1

it use the formula sin (a+b) = (sin a) (cos b) + (cos a) (sin b)
if a=b, then you'll get the formula of sin 2x

Answer 2

I like this guy's style. Most people complain when they are shown how a formula is derived. This guy actually demands (in a polite way :P) the method of derivation. You're going to become a fine engineer when you graduate. ;)

Answer 3

Thanks for the encouragement Uber: i need it right now.

Answer 4

chihirose and estudier thanks kindly
so, because a=x and b=x, i'm going to get the result sin(2x) from the above formula, right?

Answer 5

estudier: thanks for the geometrical representation: i'm going to replicate it directly using geogebra: great input, thx

Answer 6

Don't know if you can do it yourself from that diagram...

Answer 7

will fool around with the equation cited by Chihirosaleaf

Answer 8

OK, but that's another "fundamental trig identity" :-)

Answer 9

There aother methods as well. just depends how fundamental u want to get...

Answer 10

it really depends on where you start doesn't it? if you are allowed to use the double angle" formula, then since \(2\alpha=\alpha +\alpha\) you can use it, but that begs the question

if you have a grasp of the exponential form of complex numbers, then it is a trivial consequence of the laws of exponents

Answer 11

Dang it... now *I'm* curious. :P

Let's see...

sin (a+b) = (sin a) (cos b) + (cos a) (sin b)

sin (x + x) = (sinx)(cosx) + (cosx)(sinx)
sin(2x) = (sinx)(cosx) + (sinx)(cosx)
sin(2x) = (sinxcosx)(1+1)
sin(2x) = 2(sinxcosx)

Oh hey. That wasn't so bad! Granted, I do have to admit I stared at this step:

sin(2x) = (sinx)(cosx) + (sinx)(cosx)

For waaayyyy too long before I realized it factors into (sinxcosx)(1+1).

Lolz.

Answer 12

estudier, I don't mind memorizing stuff; but I was hoping for an intuitive pathway from (have to finish this later: just got a call to go and rescue a red-bellied black snake from a house around the corner: wildlife is another passion of mine: especially snakes ) thanks one and all! i'll be back!

Answer 13

big respect uber! and many thanks, will read when i get back from rescue!

Answer 14

The easiest and fastest method is via complex exponents as satellite said (if u know that).

Answer 15

No problemo; I myself wish to thank both you and chihiro for motivating me to try; this was a nifty experience. Heheh.

Answer 16

big respect uber! and many thanks, will read when i get back from rescue!

Answer 17

woops, already said that: meant to say: uber gets best response from me coz it meshes so perfectly with my current need to be: 1. instructed, 2. reassured that I'm not a dunce and 3. reminded how much fun maths can be, which is VERY motivating

Answer 18

Satellite: not covered complex numbers as yet: still trying to grasp the principles of differential calc before a precipitous dash to the integral (way too much too quick for my liking) Next year, over the break, I can take a breather and try to deepen and hone the skills I've only just played with once or twice throughtout the course so far: like the trig identities for instance; like the quotient rule for another.... ah, so much to learn
This place is very, very cool.
Thanks everyone who contributed.
PS the snake wasn't catchable when i got there; lady has to spend the night with it inside the house, somewhere, but then, she has a door open in the sub-tropics here, all night! go figure

Answer 19

Thanks, godfrey. I owe chihiro a medal as well for giving us the formula (and I would have medal'd the rest of you guys, too, but unfortunately they seem to have placed a one medal limit in the year that I've been gone from here).