Convergence of Fourier series - strange graph in proof

Question

I am reading a text that states the following related to convergence of Fourier series:

$$g_K(x) = > \frac{1}{2\pi}\int_{x-2\pi}^x\frac{\sin\left[\left(K+\frac{1}{2}\right)t\right]}{\sin\left(\frac{1}{2}t\right)}f(x-t)\,dt$$

The kernel $\sin(K+\frac{1}{2})t/\sin(\frac{1}{2}t)$ of the previous integral is plotted for several values of $K$ in the following figure.

However, the caption of the corresponding figure says "a plot of $\sin[(K+\frac{1}{2})t]/[(K+\frac{1}{2})t]$", and indeed, the plots are of

$$\frac{\sin[(K+\frac{1}{2})t]}{(K+\frac{1}{2})t}$$

for several $K$. Is the author then plotting a completely wrong function, or could I be missing something? It seems too wrong to simply be a typo. Thanks all.

Answer 1

Indeed, the combination of this text

and this figure

is clearly a mistake. It's not uncommon for a student to be charged with making plots for a book, and Artem gave a plausible guess for how a mistake like this could happen.