To solve for $x$, we are given a right triangle with:

• The side opposite to $x^\circ$ = 29
• The hypotenuse = 59

We can use the sine function, defined as: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

Substitute the known values: $\sin(x) = \frac{29}{59}$

Calculate the ratio: $\sin(x) = 0.4915254237$

Now, find the angle $x$ by taking the inverse sine ($\sin^{-1}$) of the ratio: $x = \sin^{-1}(0.4915254237)$

Using a calculator: $x \approx 29.4^\circ$

Final Answer:

$x \approx 29.4^\circ$

Would you like a detailed explanation of the trigonometric steps or related concepts?

Related Questions:

1. What are the other trigonometric ratios (cosine and tangent) for the given triangle?
2. How would you calculate the angle at $Q$ in the triangle?
3. What is the length of side $PQ$ using the Pythagorean theorem?
4. Why do we use the sine function instead of cosine or tangent in this problem?
5. What does it mean to round to the nearest tenth, and why is it important?

Tip:

When solving right triangles, always double-check which side is the opposite, adjacent, or hypotenuse relative to the angle you are working with!