To solve this problem, we need to find the length of the side $x$ in a right triangle where we are given the length of the adjacent side (19 units) and the angle $46^\circ$.

Step-by-Step Solution

1. Identify the trigonometric relationship: Since we know the angle and the adjacent side, and we're trying to find the hypotenuse ($x$), we use the cosine function: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

2. Set up the equation: Plugging in the values: $\cos(46^\circ) = \frac{19}{x}$

3. Solve for $x$: Rearrange to isolate $x$: $x = \frac{19}{\cos(46^\circ)}$

4. Calculate $x$: Using a calculator to find $\cos(46^\circ) \approx 0.6946$: $x = \frac{19}{0.6946} \approx 27.4$

Answer

The length of the missing side $x$ is approximately 27.4 units.

Would you like a more detailed breakdown or have any other questions?

Here are five related questions you could explore:

1. How would the calculation change if the angle was 30 degrees instead of 46 degrees?
2. What is the value of the opposite side if we know the hypotenuse?
3. How can we find the angle if we only know two sides of the triangle?
4. What trigonometric function would we use if we knew the opposite side and wanted to find the hypotenuse?
5. How would this solution change if we were given the opposite side instead of the adjacent side?

Tip: Always double-check your calculator is in the correct mode (degrees vs. radians) when working with angles!