The image shows the solution to a triangle problem, and you're using the Pythagorean theorem and trigonometric functions. Let’s walk through it:

Part 1: Pythagorean Theorem for $x$

You first use the Pythagorean theorem to solve for side $x$. The equation used is: $a^2 + b^2 = c^2$ Substituting the values from your diagram: $6^2 + 2^2 = c^2$ $36 + 4 = c^2 \quad \Rightarrow \quad c^2 = 40 \quad \Rightarrow \quad c = \sqrt{40} \approx 6.3$ This is correct! $x = 6.3$.

Part 2: Solving for $y$

You use the sine function to find angle $y$ in the right triangle. The formula is: $\sin(y) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5.56}{9}$ Then, you correctly use the inverse sine function: $y = \sin^{-1}\left(\frac{5.56}{9}\right) \approx 38.1^\circ$ This calculation is also correct.

Conclusion

The steps you followed are accurate for both using the Pythagorean theorem and trigonometry. Well done!

Let me know if you need further clarification or have any additional questions.

Here are some related questions you might find interesting:

1. How do you use the cosine rule to solve for unknown sides or angles in non-right triangles?
2. What would happen if you had an obtuse triangle in this situation?
3. How would you solve a triangle using the Law of Sines instead of the Law of Cosines?
4. Can you explain how to convert between degrees and radians for trigonometric functions?
5. What are some common mistakes to avoid when applying the Pythagorean theorem?

Tip: When solving trigonometric problems, always check that you’re using the right triangle relationships based on the given information (opposite, adjacent, or hypotenuse).