Question

find 3 coeff. b1, b2, b3 E of R3 such that:
sin(t)cos(2t) = b1sin(t) + b2sin(2t) + b3sin(3t).
i just need a clue of the formula i need to use. thx alot.

Answer 1

It looks like you'd need to check out one of the several double angle formulas for cos(2t) and it should become pretty simple at that point.

Answer 2

that's what i thought, but it's a linear algebra class so I think it's a little tougher than that.

Answer 3

waaaa never mind i think it's that simple.

Answer 4

So, I'm assuming you're being asked to prove that sin(t)cos(2t) is in the span of {sin(t), sin(2t), sin(3t)} or something similar?

Yeah, I think in this case it really is going to be simple. Sometimes with Linear Algebra problems are almost so simple that they look hard! :)

Answer 5

Of course the real problem is that then there is a problem that looks really simple, but ends up being really difficult and it's not always clear which is which. :)

Answer 6

(sin t)(cos (2t)) = (sin t)(1 – 2(sin t)^2) = sin t – 2 (sin t)^3.

sin(3t) =sin (2t + t) = sin (2t) cos (t) + cos (2t) sin(t) = … = 3sin t – 4(sin t)^3.
sin(2t) = 2(sin t)(cos t).
sin (t) = sin (t).

So now notice the Left Hand Side has no cosines.
This suggests letting b2 = 0.
Notice that the Left Hand Side has coefficient -2 on the cubic term.
This suggests letting b3 = ½.
But if you make b3 equal to 1/2, then it contributes 3*(1/2) to the coefficient of sin t.
To compensate, we should let b1 be -1/2.

So (-1/2, 0, 1/2) is the solution.

Answer 7

Or is a solution, I'm not sure that it's unique.

Answer 8

thanks a lot i verified my answer and it's for sure correct.

Answer 9

cool