Task B-3.2. Calculate the area of a rhombus whose shorter diagonal is $8 \mathrm{~cm}$ long, and the measure of the obtuse angle is $150^{\circ}$.
Solution.

![](https://cdn.mathpix.com/cropped/2024_05_30_7c0a9615fb5cd46ea1c5g-12.jpg?height=393&width=1111&top_left_y=1603&top_left_x=472)

From the right triangles $B N D$ and $A N D$, we get

$$
\begin{aligned}
v & =8 \cdot \sin 75^{\circ} \\
a & =\frac{v}{\sin 30^{\circ}}=\frac{8 \cdot \sin 75^{\circ}}{\sin 30^{\circ}} \\
P & =a \cdot v=\frac{64 \cdot \sin ^{2} 75^{\circ}}{\sin 30^{\circ}} \\
P & =128 \cdot \frac{1-\cos 150^{\circ}}{2}=64 \cdot\left(1+\frac{\sqrt{3}}{2}\right)=64+32 \cdot \sqrt{3} \mathrm{~cm}^{2}
\end{aligned}
$$

(2 points)

Note: Any correct expression for the area of the triangle that includes trigonometric functions of angles $15^{\circ}, 75^{\circ}$ is worth 2 points. For example, $P=32 \operatorname{tg} 75^{\circ}, P=\frac{32}{\operatorname{tg} 15^{\circ}}, P=\frac{8}{\sin ^{2} 15^{\circ}}$. Calculating the exact value of the area is another 2 points.