By calculating a Taylor approximation, determine K

[Jozefina Gramatikova]

Homework Statement

Homework Equations

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The Attempt at a Solution

Can somebody explain to me how did we find the function in red? Thanks

 

[Charles Link]

I recommend you first expand this function using $\sin(\theta-\phi)=\sin(\theta) \cos(\phi)-\sin(\phi) \cos(\theta)$, and then work a Taylor series after putting in the values for $\theta$ and $\phi$.

 

[Charles Link]

Assuming you did the above step, what do you get for the Taylor expansion of $\cos(\frac{\pi}{2} x)$?

 

[Charles Link]

I'm going to give you a couple more hints on this problem, because it is really quite a neat one: $\\$ Let $f(x)=\cos(\frac{\pi}{2}x )$. This can be written as $f(x)=\cos(bx)$ where $b=\frac{\pi}{2}$. Now, by the chain rule, $f'(x)=-b \sin(bx)$, and $f''(x)=-b^2 \cos(bx)$. The Taylor series $f(x)=f(0)+f'(0)(x-0)+f''(0) \frac{(x-0)^2}{2!}+...$ $\\$ I gave you a couple of the terms here to get you started, but this problem actually will require even the 4th derivative. (Compute these first couple of terms and you will see why).