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We have a right triangle with angle \( \angle Z = 58^\circ \), side \( TZ = 6.7 \), and side \( YZ = x \) that we need to solve for.

Since this is a right triangle, we can use trigonometric ratios to find the unknown side \( x \). Specifically, we can use the sine function, since we know the angle \( \angle Z \) and the length of the hypotenuse \( TZ \).

The sine of an angle in a right triangle is given by:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

For this triangle:
- The opposite side to \( \angle Z \) is \( YZ = x \),
- The hypotenuse is \( TZ = 6.7 \),
- The angle is \( 58^\circ \).

Substitute the known values into the sine function:

\[
\sin(58^\circ) = \frac{x}{6.7}
\]

Now, solve for \( x \):

\[
x = 6.7 \times \sin(58^\circ)
\]

Using a calculator to find \( \sin(58^\circ) \):

\[
x = 6.7 \times 0.8480 \approx 5.7
\]

Thus, \( x \approx 5.7 \). 

Would you like further details or have any questions?

Here are five related questions to expand on this:
1. How can you use the cosine function to find the length of side \( XY \)?
2. What would happen if you were given angle \( \angle T \) instead of \( \angle Z \)?
3. How can you check your answer using the Pythagorean theorem?
4. What is the definition of the tangent ratio, and how might it apply here?
5. How can trigonometry be used to find angles in non-right triangles?

**Tip:** Always ensure your calculator is in degree mode when working with angles measured in degrees.

Solve for x in a right triangle with angle Z = 58°, side TZ = 6.7, and side YZ = x. Round to the nearest tenth if necessary.

This problem involves solving for the unknown side x of a right triangle using the sine function. Given angle Z = 58° and hypotenuse TZ = 6.7, students will learn how to apply the sine function to find the opposite side. Suitable for high school students in Grades 9-10.

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