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To find the size of angle \( YZX \) in the given right-angled triangle, we can use trigonometric functions. Specifically, we have two sides of the triangle:

- The side opposite angle \( YZX \) is \( XY = 7.5 \, \text{cm} \) (adjacent to angle X).
- The hypotenuse \( XZ = 22.3 \, \text{cm} \).

We can use the sine function because:

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

Thus, for angle \( YZX \):

\[
\sin(YZX) = \frac{7.5}{22.3}
\]

Now let's calculate \( \sin(YZX) \) and find the angle.

\[
\sin(YZX) = \frac{7.5}{22.3} \approx 0.336
\]

To find the angle \( YZX \), we use the inverse sine (arcsine) function:

\[
YZX = \sin^{-1}(0.336)
\]

Using a calculator, we get:

\[
YZX \approx 19.6^\circ
\]

Thus, the size of angle \( YZX \) is approximately **19.6°** to 3 significant figures.

Would you like a deeper explanation of the trigonometric process or any other details?

Here are 5 related questions:
1. How would you find angle \( XZY \) using the available sides?
2. What trigonometric function would you use if you had the adjacent side instead of the opposite?
3. How could the Pythagorean theorem verify the correctness of this triangle?
4. What is the cosine of angle \( YZX \)?
5. How would you calculate the area of this right triangle?

**Tip:** In right triangles, the sine, cosine, and tangent functions are powerful tools for calculating unknown angles when side lengths are known.

Find the size of angle YZX. Give your answer to 3 significant figures.

Learn how to find the size of angle YZX in a right triangle with sides 7.5 cm and 22.3 cm. This trigonometry problem demonstrates how to use the sine function to solve for an angle, suitable for students in Grades 9-10.

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