Question

How can I simplify sin^5x? I need to write it in terms of sinx, sin(3x) and sin(5x).

Answer 1

Hahaha I know how to use wolframalpha! I don't how to get there :P

Answer 2

Use Euler's formula. You know it?

Answer 3

No, I'll look into it though and post back.

Answer 4

\[\Im((e^{i\theta})^5)=\Im(e^{i5\theta})\]

Answer 5

Definitely don't know that! This is a linear algebra course, the original question is find the coordinate vector for sin^5(x) from the basis {sinx,sin3x,sin5x}.

So far, I've been trying to use addition rules to simplify them. sin(3x)=(sin2x+x) etc.

But it's taking forever, there must be a better way!

Answer 6

\[e^{ix}=\cos(x)+\sqrt{-1}\sin(x)\]
\[(e^{ix})^2=(\cos(x)+\sqrt{-1}\sin(x))^2=cos^2(x)-sin^2(x)+2\sqrt{-1}cos(x)sin(x)\]
\[(e^{ix})^2=e^{i2x}=cos(2x)+\sqrt{-1}sin(2x)\]Use the fact that if you have
\[f(x)+\sqrt{-1}g(x)=h(x)+\sqrt{-1}p(x)\]Then this MUST be true:
\[f(x)=h(x)\] and \[g(x)=p(x)\]

Answer 7

Use for\[e^{i5x}=(e^{ix})^5\]The \[\Im\]bit just means 'remove all the things that do not have a coefficient of \[\sqrt{-1}\] in front of them.

Answer 8

\[\sqrt{-1}=i=\text{ imaginary number } \]
My method may look tricky, but it's MUCH simpler

Answer 9

http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/

Answer 10

@henpen I think the equations you are using are beyond the level o math @blufoot is at I believe he is suppose to be using the base trig identites

Answer 11

*of

Answer 12

Fair enough, but it's always useful to know where the base identities come from. You help- I need to sleep.

Answer 13

Blufoot, you're on the right track, use sin(a+b)=sinacosb+sinbcosa

Answer 14

Hey that's ok this is interesting! I remember learning a bit about imaginary numbers back in high school, but I don't see how it leads me to solving for sin(5x) :P

I guess I'll just have to use the identities :(

sin(3x) is easy, but sin(5x) takes forever because it's sin(3x+2x) which is cos(2x)sin(3x)+sin(2x)cos(3x)... So I need cos(3x) also and a ton of simplifying... I guess I'll give in to wolframalpha!

Answer 15

wait is this
\[\sin ^{5}x\]
or
\[\sin(5x)\]

Answer 16

Oh, either one, really. I'm trying to write sin^5x in terms of the other 3. I'm not sure if sin^5x is an easier one to simplify? Couldn't find a rule for that one...

Answer 17

there is a rule that may help with the load if you can figure out how to deal with the cos(x) its
\[\sin ^{2}(x)=\frac{ 1-\cos(2x) }{ 2 }\]

Answer 18

which I would think means
\[\sin ^{5}(x)=\frac{ 1-\cos(5x) }{ 5 }\]
but I could also be horribly mistaken in that conclusion

Answer 19

I'm afraid you are :(

Answer 20

I am sorry but I actually have got to go bye