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To solve for angle \(x\) in the given right-angled triangle using trigonometry, we can use the **tangent function** since we are given the opposite side (3 cm) and the adjacent side (8 cm) to angle \(x\).

The formula for the tangent of an angle in a right-angled triangle is:

\[
\tan(x) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{8}
\]

Now, to find \(x\), we take the inverse tangent (\(\tan^{-1}\)) of the ratio:

\[
x = \tan^{-1} \left( \frac{3}{8} \right)
\]

Let me calculate that for you.The size of angle \(x\) is approximately \(20.56^\circ\).

Would you like more details or have any questions about the solution?

Here are 5 related questions for further practice:
1. What is the sine of angle \(x\) in this triangle?
2. How would you find the hypotenuse of the triangle?
3. Can you calculate the area of the triangle?
4. What would the angle \(x\) be if the opposite side was 5 cm instead of 3 cm?
5. How would you use the cosine function to find another angle in the triangle?

**Tip:** When dealing with right-angled triangles, always identify the sides relative to the angle in question (opposite, adjacent, and hypotenuse) before choosing the correct trigonometric ratio.

Shown below is a right-angled triangle with sides of 3 cm and 8 cm. Use trigonometry to work out the size of angle x.

Learn how to find the size of angle x in a right-angled triangle with sides 3 cm and 8 cm using trigonometry. This problem demonstrates the use of the tangent function and inverse tangent to calculate angles. Suitable for students in Grades 8-10.

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