1. ## Trigonometry simplification

Can anybody help in simplifying the trigonometric equation under Question 2 in the attached document?

Thanks

2. Originally Posted by a_shyam41
Can anybody help in simplifying the trigonometric equation under Question 2 in the attached document?

Thanks

$S(x)=\sum_{n=1}^{\infty}a_n \sin \left(\frac{(n+1/2)x}{h}\right)$

where:

$
a_n=\frac{\cos(n\pi)\sqrt{\beta}}{\sqrt{3+\frac{n^ 4}{a^4}}}
$

First of all observe that $\cos(n\pi)=(-1)^n$ and that $\sqrt{\beta}$ is a constant and so can be taken outside the summation.

$
a_n=a^4 \sqrt{\beta}\frac{(-1)^n}{\sqrt{3a^4+n^4}}
$

$
S(x)=a^4 \sqrt{\beta}\ \sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt{3a^4+n^4}} \sin \left(\frac{(n+1/2)x}{h}\right)
$

Next thought is: can we turn this into a Fourier series (that we recognise)? I don't have the time right now but may return to this.

CB