We are given a right triangle, where:

• $\angle D = 22^\circ$
• $DE = 16$ (the length of the side opposite the angle)
• $x = FE$ (the hypotenuse, which we need to find)

Since we have an angle and the opposite side, and we need to find the hypotenuse, we can use the sine function.

$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

For this triangle:

$\sin(22^\circ) = \frac{16}{x}$

Now solve for $x$:

$x = \frac{16}{\sin(22^\circ)}$

First, calculate $\sin(22^\circ)$ (using a calculator):

$\sin(22^\circ) \approx 0.3746$

Now solve for $x$:

$x = \frac{16}{0.3746} \approx 42.7$

So, $x \approx 42.7$.

Would you like a detailed breakdown of any part of the solution?

Here are 5 related questions to explore further:

1. How do we calculate the adjacent side using trigonometry in a right triangle?
2. What is the relationship between the other trigonometric functions and this triangle?
3. How can we verify the result using the Pythagorean theorem?
4. What is the tangent of $22^\circ$, and how would it relate to this triangle?
5. How would the calculation change if the given angle were different?

Tip: Always check your calculator settings to ensure you're working in degrees (not radians) when dealing with angles like $22^\circ$.