To solve for $x$ in this right triangle $\triangle WTY$, we can use trigonometric ratios. Here's the solution process:

Step 1: Identify the given information

1. The triangle has:

   • $\angle W = 48^\circ$,
   • $WT = 52$ (adjacent side to $\angle W$),
   • $TY$ is the opposite side of $\angle W$, denoted $x$,
   • $\angle T = 90^\circ$.

2. We need to solve for $x$, which is opposite $\angle W$.

Step 2: Use the tangent ratio

The tangent of an angle in a right triangle is defined as: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

For $\angle W = 48^\circ$, we substitute the known values: $\tan(48^\circ) = \frac{x}{52}$

Step 3: Solve for $x$

Rearranging the equation to isolate $x$: $x = 52 \cdot \tan(48^\circ)$

Using a calculator to find $\tan(48^\circ)$: $\tan(48^\circ) \approx 1.1106$

Substitute this value into the equation: $x = 52 \cdot 1.1106$

$x \approx 57.7$

Final Answer:

$x \approx 57.7$ (Rounded to the nearest tenth)

Do you have any questions or need further details on this solution?

Related Questions:

1. How do you calculate other side lengths in this triangle using trigonometry?
2. What is the hypotenuse length $WY$ of $\triangle WTY$?
3. Can you solve for $x$ using the sine function instead?
4. How do you confirm the solution using the Pythagorean theorem?
5. What are the steps to find all angles in a right triangle?

Tip:

When solving right triangles, always identify which trigonometric ratio (sine, cosine, or tangent) is most appropriate based on the given sides and angles.