fourier series of $f(x)=\sin(25t)\cos(t)$

Question

I got a trouble with this .

$B(n)$ was $\displaystyle \frac{2\pi}{25} \int_0^{\pi/25} \sin(25t) \cos(t) \sin \left( \frac{25nt}2 \right) \, dt$

and semi -result was $\displaystyle \frac{2\pi}{25} \int_0^{\pi/25} \left( \frac{\sin(26t)+\sin(24t)} 2 \right) \sin\left(\frac{25nt} 2 \right) \, dt$.

how can degrade this perfectly? it keeps teasing me...

Answer 1

In this case I wouldn't go to calculate the coefficients with the projection. Instead, I would use the formula for $\cos{a}\sin{b} = \frac{1}{2}\left(\sin(a+b) + \sin(b-a)\right)$. Use this and the uniqueness of representation.