To find the area of a triangle when two sides and the included angle are given, we use the formula:

$\text{Area} = \frac{1}{2}ab\sin(C)$

Given:

• $a = 45.5$
• $b = 35.8$
• $C = 133.4^\circ$

Step 1: Convert angle $C$ to radians (for calculations involving trigonometric functions)

$C_{\text{radians}} = C_{\text{degrees}} \times \frac{\pi}{180} = 133.4 \times \frac{\pi}{180} \approx 2.328 \text{ radians}$

Step 2: Calculate $\sin(C)$

$\sin(C) = \sin(2.328) \approx 0.728$

Step 3: Substitute into the area formula

$\text{Area} = \frac{1}{2} \cdot 45.5 \cdot 35.8 \cdot 0.728$

$\text{Area} \approx \frac{1}{2} \cdot 45.5 \cdot 35.8 \cdot 0.728 \approx 592.65 \, \text{square units}$

Final Answer:

The area of $\triangle ABC$ is approximately 592.65 square units.

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Do you want a breakdown of these steps or further clarification?

Expanding Questions:

1. How is the sine function used to find areas involving non-right triangles?
2. Why is it necessary to convert degrees to radians in this formula?
3. What is the significance of the included angle in this calculation?
4. How would the calculation differ if $C$ was a right angle?
5. Can we use Heron's formula for this problem instead? Why or why not?

Tip:

Always ensure your calculator is set to the correct mode (degrees or radians) when performing trigonometric calculations!