To solve for $x$ in this right triangle, we will use trigonometric functions, specifically the sine function.

Step 1: Understand the problem

In the triangle:

• The given angle is $\angle C = 46^\circ$.
• The side opposite $\angle C$ is $BC = 44$.
• The hypotenuse is $AB = x$.

The sine function is defined as: $\sin(\text{angle}) = \frac{\text{opposite}}{\text{hypotenuse}}.$

Step 2: Apply the sine function

For $\angle C = 46^\circ$: $\sin(46^\circ) = \frac{BC}{AB}.$

Substitute the values: $\sin(46^\circ) = \frac{44}{x}.$

Step 3: Solve for $x$

Rearrange the equation: $x = \frac{44}{\sin(46^\circ)}.$

Step 4: Compute the value

Using a calculator: $\sin(46^\circ) \approx 0.7193.$

Now calculate $x$: $x = \frac{44}{0.7193} \approx 61.2.$

Final Answer:

$x \approx 61.2$

Let me know if you want further clarification or additional details!

Follow-up Questions:

1. What other trigonometric functions could we use if we were given different sides?
2. How would the calculation change if the angle or opposite side was different?
3. What are the properties of a right triangle that help in solving these problems?
4. How does the cosine function differ from sine in right triangle calculations?
5. What are some common real-world applications of trigonometric functions?

Tip:

Always ensure your calculator is in the correct mode (degrees or radians) based on the given problem!