B3. The function is given by $f(x)=a \cdot \cos \frac{x}{2}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_8d4f73ada7862f989e67g-09.jpg?height=511&width=531&top_left_y=273&top_left_x=1391)

a) Determine $a$ so that the graph of the function passes through the point $A\left(\frac{\pi}{2}, \sqrt{2}\right)$.
b) For $a=2$, calculate the zeros of the function and sketch its graph on the interval $[-2 \pi, 2 \pi]$.
B3. Considering that the graph of the function passes through point $A$, we have $\sqrt{2}=a \cdot \cos \frac{\pi}{4}$. From this, we get $a=2$. We then calculate the zeros of the function $f(x)=2 \cos \frac{x}{2}$. These are: $x_{k}=\pi+2 k \pi ; k \in \mathbb{Z}$. We draw the graph of the function on the interval $[-2 \pi, 2 \pi]$.

![](https://cdn.mathpix.com/cropped/2024_06_07_8d4f73ada7862f989e67g-16.jpg?height=371&width=985&top_left_y=1091&top_left_x=273)

![](https://cdn.mathpix.com/cropped/2024_06_07_8d4f73ada7862f989e67g-16.jpg?height=68&width=1641&top_left_y=1488&top_left_x=276)

![](https://cdn.mathpix.com/cropped/2024_06_07_8d4f73ada7862f989e67g-16.jpg?height=48&width=1639&top_left_y=1552&top_left_x=274)

Calculated zeros $x_{k}=\pi+2 k \pi ; k \in \mathbb{Z} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

Correctly drawn graph . .................................................................................

NOTE: The amplitude must be considered on the graph, and two zeros must be marked.