3.82. The ratio of the area of the diagonal section of a regular quadrilateral pyramid to the area of its base is $k$. Find the cosine of the plane angle at the vertex of the pyramid.
3.82. Given that $P A B C D$ is a regular quadrilateral pyramid, $S_{\triangle P B D}: S_{A B C D}=k$ (Fig. 3.83); we need to find $\cos \angle D P C$. Let $D C=a$ and $O P \doteq h$; then $\frac{0.5 a \sqrt{2} h}{a^{2}}=k, h=a k \sqrt{2}$. From $\triangle P O D$ we find $D P^{2}=2 a^{2} k^{2}+\frac{a^{2}}{2}=\frac{a^{2}}{2}\left(4 k^{2}+1\right)$, and in $\triangle D P C$ by the cosine theorem we have

$$
a^{2}=a^{2}\left(4 k^{2}+1\right)-a^{2}\left(4 k^{2}+1\right) \cdot \cos \angle D P C
$$

from which $\cos \angle D P C=\frac{4 k^{2}}{4 k^{2}+1}$.

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-553.jpg?height=416&width=512&top_left_y=386&top_left_x=73)

Fig. 3.83

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-553.jpg?height=475&width=527&top_left_y=364&top_left_x=629)

Fig. 3.84