To solve for $x$ in the given right triangle $TVW$, we will use trigonometric ratios. Here’s the step-by-step process:

Triangle Analysis:

1. Given:

   • $\angle T = 61^\circ$,
   • Adjacent side $VW = 2.6$ (with respect to $\angle T$),
   • Hypotenuse $x$ is what we need to find.

2. Trigonometric Ratio:

   • The cosine function relates the adjacent side and the hypotenuse: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}.$ Here, $\theta = 61^\circ$, adjacent = $2.6$, and hypotenuse = $x$.

Step-by-Step Solution:

1. Write the cosine equation: $\cos(61^\circ) = \frac{2.6}{x}.$

2. Solve for $x$ by isolating it: $x = \frac{2.6}{\cos(61^\circ)}.$

3. Find $\cos(61^\circ)$ using a calculator: $\cos(61^\circ) \approx 0.4848.$

4. Substitute this value into the equation: $x = \frac{2.6}{0.4848}.$

5. Perform the division: $x \approx 5.4.$

Final Answer:

The value of $x$, rounded to the nearest tenth, is: $\boxed{5.4}$

Let me know if you need further explanation or details!

Related Questions:

1. How would you solve for $VW$ if the hypotenuse and angle were different?
2. Can you explain the sine ratio in this triangle?
3. How do you determine the third angle in a right triangle?
4. What are the steps to use the tangent function instead?
5. How do trigonometric ratios differ from the Pythagorean Theorem?

Tip:

Always identify the opposite, adjacent, and hypotenuse sides relative to the angle you are working with to pick the correct trigonometric ratio.