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

- anonymous

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

- schrodinger

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- anonymous

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

- anonymous

Use Euler's formula. You know it?

- anonymous

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

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- anonymous

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

- anonymous

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!

- anonymous

\[e^{ix}=\cos(x)+\sqrt{-1}\sin(x)\] \[(e^{ix})^2=(\cos(x)+\sqrt{-1}\sin(x))^2=cos^2(x)-sin^2(x)+2\sqrt{-1}cos(x)sin(x)\] \[(e^{ix})^2=e^{i2x}=cos(2x)+\sqrt{-1}sin(2x)\]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 \[g(x)=p(x)\]

- anonymous

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.

- anonymous

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

- anonymous

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

- anonymous

@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

- anonymous

*of

- anonymous

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

- anonymous

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

- anonymous

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!

- anonymous

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

- anonymous

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...

- anonymous

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

- anonymous

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

- anonymous

I'm afraid you are :(

- anonymous

I am sorry but I actually have got to go bye

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