The image shows a right-angled triangle with the angle of $72^\circ$ at one vertex and a given side of 2.2 m (opposite the $72^\circ$ angle). We are tasked with finding the length of the adjacent side, $b$, using trigonometry.

To solve for $b$, we can use the tangent function, which relates the angle in a right triangle to the ratio of the opposite and adjacent sides:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Here, $\theta = 72^\circ$, the opposite side is 2.2 m, and the adjacent side is $b$. Rearranging the equation to solve for $b$:

$b = \frac{\text{opposite}}{\tan(\theta)} = \frac{2.2}{\tan(72^\circ)}$

Let me calculate the value of $b$.The length of side $b$ is approximately $0.715$ meters, rounded to three significant figures.

Would you like more details or have any questions about this solution?

Here are five related questions to expand on this topic:

1. How would the calculation change if the given angle was smaller, say $60^\circ$?
2. Can you calculate the hypotenuse of this triangle using trigonometric functions?
3. What would happen if the opposite side were doubled while keeping the angle $72^\circ$?
4. How do you solve for side $a$ (the vertical side)?
5. How can you use the sine or cosine rule to verify this result?

Tip: Remember that the tangent function is only defined for angles between $0^\circ$ and $90^\circ$ and not at $90^\circ$ itself.