To calculate the angles between the adjacent sides of a rhombus when the lengths of the diagonals are given, we can use trigonometric relationships. Here's the process:

Step 1: Understand the problem

A rhombus has equal sides, and its diagonals bisect each other at right angles (90°). If the diagonals are $d_1 = 22 \, \text{cm}$ and $d_2 = 15 \, \text{cm}$, then the diagonals divide the rhombus into four right-angled triangles.

Step 2: Use trigonometry

Consider one of these right triangles formed by half of the diagonals. The lengths of the two legs of this triangle are: $\text{Half of } d_1 = \frac{22}{2} = 11 \, \text{cm}$ $\text{Half of } d_2 = \frac{15}{2} = 7.5 \, \text{cm}$

We need to find the angle between the adjacent sides of the rhombus, which we'll denote as $\theta$.

In a right triangle, the tangent of the angle is given by: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{7.5}{11}$

Step 3: Calculate the angle

$\theta = \tan^{-1} \left(\frac{7.5}{11}\right)$ Let's calculate this angle.

Step 4: Double the angle

Since this angle $\theta$ is only half of the angle between the adjacent sides of the rhombus, the angle between the adjacent sides is $2\theta$.

Let's now calculate this.The angle between the adjacent sides of the rhombus is approximately 68 degrees and 34 minutes.

Would you like any further details or have any questions?

Here are five related questions for further exploration:

1. How would the area of the rhombus be calculated given the diagonals?
2. What are the properties of the diagonals in any rhombus?
3. How would the angles change if the diagonals were of equal length?
4. Can you calculate the side length of the rhombus using the given diagonals?
5. How do the properties of a rhombus relate to those of other quadrilaterals like squares and rectangles?

Tip: Understanding the relationship between the sides and angles of polygons through trigonometry can help solve complex geometry problems more easily.