To find the length $k$, we can use trigonometry, specifically the sine function, since $k$ is the hypotenuse and the given angle is opposite to the side of 7.7 cm.

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

From the triangle: $\sin(36^\circ) = \frac{7.7}{k}$

Rearranging for $k$: $k = \frac{7.7}{\sin(36^\circ)}$

Step-by-Step Solution:

1. Calculate $\sin(36^\circ)$: $\sin(36^\circ) \approx 0.5878$

2. Substitute into the equation for $k$: $k = \frac{7.7}{0.5878}$

3. Compute the result: $k \approx 13.1 \, \text{cm (to 1 decimal place)}.$

Final Answer:

$k \approx 13.1 \, \text{cm}$

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Related Questions:

1. What is the formula for the cosine function, and when should it be used in a triangle?
2. How would you calculate the length of the side adjacent to the angle $36^\circ$?
3. What are the steps to verify this calculation using the Pythagorean theorem?
4. How can you calculate the area of this triangle using $k$?
5. What happens to the value of $k$ if the angle changes to $45^\circ$?

Tip:

Always ensure your calculator is in degree mode when working with angles measured in degrees.