To solve for angle $x$ in the given right triangle, we can use trigonometric functions. The sides provided are the opposite side (length $8$) and the adjacent side (length $6$) relative to angle $x$.

Step-by-Step Solution:

1. Identify the trigonometric function to use:

   • Sine function: $\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}}$
   • Cosine function: $\cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}}$
   • Tangent function: $\tan(x) = \frac{\text{opposite}}{\text{adjacent}}$

   Since you have the lengths of the opposite side (8) and the adjacent side (6), the appropriate function to use is Tangent.

2. Apply the tangent function: $\tan(x) = \frac{\text{opposite}}{\text{adjacent}} = \frac{8}{6} = \frac{4}{3}$

3. Solve for $x$: $x = \tan^{-1}\left(\frac{4}{3}\right)$ Using a calculator or trigonometric table: $x \approx 53.13^\circ$

Answer:

The correct function to use is C. Tangent, and the value of $x$ is approximately $53.13^\circ$.

Would you like more details on how to calculate the arctangent or have any other questions?

Related Questions:

1. What are the exact values for sine and cosine of this angle $x$?
2. How would you solve for the hypotenuse using the given side lengths?
3. What is the relationship between sine, cosine, and tangent in a right triangle?
4. How can you verify the angle using the sine or cosine function?
5. How do you convert this angle from degrees to radians?

Tip: In trigonometry, it's important to always identify which sides of the triangle you are dealing with (opposite, adjacent, hypotenuse) before selecting a trigonometric function.