Whats a non-trivial linear combination of these functions?

[warfreak131]

I have to find a non-trivial, linear combination of the following functions that vanishes identically.

In other words

C₁f + C₂g + C₃h = 0

Where C₁, C₂, and C₃ are all constant, and cannot all = 0.

f(x)=17
g(x)=2Sin²(x)
h(x)=3Cos²(x)

I figure C₁ = 0, because there's really no constant relation between the trig functions and 17.

That means that C₂2Sin²(x)=-C₃3Cos²(x)

I need help finding C₂ and C₃. I've already tried substituting with trig identities, but I am getting nothing as of now.

 

[tiny-tim]

hi warfreak131!

c'mon … think! … what's the most famous equation involving cos² and sin² ?

 

[warfreak131]

hi warfreak131!

c'mon … think! … what's the most famous equation involving cos² and sin² ?

I know that sin²x + cos²x = 1, and I've tried that, but I am still not getting it

C₂2Sin²(x)=-C₃3Cos²(x)

C₂2(1-cos²(x))=-C₃3Cos²(x)

C₂(2-2cos²(x))=-C₃3cos²(x)

but this doesn't simplify into a constant solution, as far as i worked it out

 

[tiny-tim]

I know that sin²x + cos²x = 1 …

ok, so what = 17?

get some sleep! :zzz:​

 

[warfreak131]

ok, so what = 17?

get some sleep! :zzz:​

17sin²x+17cos²x?

so C1 = 17Sin²x + 17Cos²x

then just apply another coefficient to 2Sin²x, and 3Cos²x to make the new coefficient equal 17?

 

[tiny-tim]

yup!

g'night! :zzz:​

 

[warfreak131]

awesome thanks