Question

Rewrite sin^4(x) so that it involves only the first power of cosine.

Answer 1

@Spacelimbus can you help maybe?

Answer 2

I am not quite sure about the question itself, do you know what you're currently studying? Double angular formulas for trigonometric identities? Or have you heard of Demoivre's theorem?

Answer 3

i am doing trig identities, with double angle identities yeah

Answer 4

and have you had an introduction to complex numbers? In special Demoivre?

Answer 5

no i've not heard of demoivre

Answer 6

Because, although I understand the nature of this problem, all I could do would give you some formulas which you would have to take as granted, there is not much more to it unfortunately :(

On the other hand, if you know a bit about complex numbers then you could derive a formula yourself using \[\large \text{cis}(\alpha)= \cos \alpha + i \sin \alpha \implies \text{cis}(\alpha)^4= \text{cis}(4 \alpha)= \cos(4 \alpha) + i \sin(4 \alpha) \]

Answer 7

is there any way to do it with just the double angle or squared identities ?

Answer 8

Give me a moment to think, maybe I will manage to come up with something, you know for instance that: \[\large \sin (x) = \frac{1}{2}(1- \cos(2x)) \]

Answer 9

Sorry it's supposed to be \(\sin ^2 (x)\)

Answer 10

so you can square this identity and then try to come up with a double angular formula on the right hand side of it,

Answer 11

seems to be at least one step into the correct direction, but it might still get tedious since we want to make the \( \cos ^2(2x)\) term after quadrating vanish. Can you follow so far?

Answer 12

\[\large \sin ^4(x)= \left( \frac{1}{2}(1- \cos(2x) \right)^2= \frac{1}{4}(1 - 2 \cos(2x) + \cos^2(2x)) \]

Answer 13

yes i follow that

Answer 14

good, so for the very right identity, name \[\large \cos^2(2x)=1- \sin ^2(2x) \] right? And now apply the double angular theorem to \(\sin^2 (2x)\) again as above and you're done.