To solve for $c$ using a trigonometric ratio, we analyze the given triangle:

Given:

• The hypotenuse is $11$.
• The angle $60^\circ$ is given.
• $c$ is the side opposite the $60^\circ$ angle.

Steps:

The sine function relates the opposite side to the hypotenuse: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ For this triangle: $\sin(60^\circ) = \frac{c}{11}$ Using the sine value for $60^\circ$: $\sin(60^\circ) = \frac{\sqrt{3}}{2} \approx 0.866$ Substitute: $0.866 = \frac{c}{11}$ Solve for $c$: $c = 11 \cdot 0.866 = 9.526$ Rounded to two decimal places: $c \approx 9.53$

Final Answer:

$c \approx 9.53$

Let me know if you want a more detailed explanation or further assistance! Here are some related questions you might explore:

1. How do you calculate the cosine or tangent in this triangle?
2. What is the adjacent side length of the triangle?
3. How can you verify the solution using the Pythagorean Theorem?
4. Why is the sine ratio appropriate here, and not cosine or tangent?
5. How does rounding affect the accuracy of trigonometric solutions?

Tip: When solving trigonometric problems, always ensure your calculator is set to the correct mode (degrees or radians).