The problem shows a right triangle with angle $\angle C = 46^\circ$, side $BC = 44$ units, and hypotenuse $AC = x$. We are asked to solve for $x$.

To solve for $x$, we can use trigonometry, specifically the cosine function, because the cosine of an angle is defined as the adjacent side divided by the hypotenuse in a right triangle.

$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

For $\theta = 46^\circ$, the adjacent side is $BC = 44$, and the hypotenuse is $x$. So,

$\cos(46^\circ) = \frac{44}{x}$

Now, solve for $x$:

$x = \frac{44}{\cos(46^\circ)}$

Using a calculator:

$\cos(46^\circ) \approx 0.6947$

$x = \frac{44}{0.6947} \approx 63.3$

Thus, the value of $x$ is approximately $63.3$.

Would you like more details on this? Here are some related questions you might explore:

1. How do you determine when to use sine, cosine, or tangent in a right triangle?
2. How does the value of the cosine function change as the angle increases?
3. What are the properties of right triangles and their trigonometric ratios?
4. Can you derive the sine rule from basic trigonometric identities?
5. How would the problem change if the given angle was $90^\circ - 46^\circ$?

Tip: Always double-check your calculator's mode to ensure it's in degrees, not radians, when dealing with angles measured in degrees!