To solve for the angle $x$ in the given right triangle, we will use trigonometric ratios. Here, we are given:

• The length of the opposite side $BC = 5$
• The length of the hypotenuse $AB = 8.4$

The trigonometric function that relates the opposite side and the hypotenuse is the sine function:

$\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AB}$

Substitute the given values:

$\sin(x) = \frac{5}{8.4}$

Calculate the value:

$\sin(x) \approx \frac{5}{8.4} \approx 0.5952$

Now, to find $x$, we take the inverse sine (also known as arcsine) of 0.5952:

$x = \sin^{-1}(0.5952)$

Using a calculator:

$x \approx \sin^{-1}(0.5952) \approx 36.5^\circ$

So, the angle $x$ is approximately $36.5^\circ$.

Would you like further details or have any questions? Here are some additional questions related to this topic:

1. How do you use the cosine function to find an angle in a right triangle?
2. What is the tangent function and how is it used in right triangles?
3. Can you find the other non-right angle in the triangle given in the problem?
4. What are some applications of trigonometric functions in real life?
5. How do you convert between degrees and radians?
6. How can you verify the accuracy of your calculated angle?
7. What is the Pythagorean theorem and how is it used in right triangles?
8. What are the sine and cosine rules for non-right triangles?

Tip: When solving trigonometric equations, always ensure your calculator is set to the correct mode (degree or radian) based on the problem requirements.