Solving $\cos(\pi(x-1))=0$

Question

I have a second derivative that I need to use to find inflection points to create a graph. The second derivative is $$f^{\prime\prime}(x)=-4\pi^2\cos(\pi(x-1))$$

So I set the equation to $0$ and solve for $x$

$$-4\pi^2\cos(\pi(x-1))=0$$

I divide by the constant $-4\pi^2$ and get

$$\cos(\pi(x-1))=0$$

But I am basically stuck at this point. I know I need to take the inverse cosine of both sides. The result I am getting is $x=3/2$, but the answer in the book is $x=1/2$, $3/2$. Can someone help me figure out how to solve the last steps of this problem?

Answer 1

Assuming you're looking within $0 \le t \le 2\pi$.

Where does $\cos t = 0$?
At $\pi/2$ and $3\pi/2$.
Set $\pi(x-1)$ equal to each of these and solve for $x$. You'll get $3/2$ and $1/2$.

Answer 2

$\cos\frac{\pi}{2} = \cos\frac{3\pi}{2}$. There is a solution in another region of the 4 quadrants.