In this problem, we are given a right triangle with sides $a = 6$ and $b = 12$, and we are tasked with finding the angle $\theta$ opposite side $a$.

To find the angle $\theta$, we can use the tangent function, which relates the opposite and adjacent sides in a right triangle:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b}$

In this case:

$\tan(\theta) = \frac{6}{12} = 0.5$

Now, we find the angle $\theta$ by taking the arctangent (inverse tangent) of 0.5:

$\theta = \tan^{-1}(0.5)$

Let's calculate the angle to one decimal place.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can calculate the angle manually. Here's how:

1. Use a scientific calculator to find the inverse tangent of 0.5: $\theta = \tan^{-1}(0.5) \approx 26.6^\circ$

If you have any further questions or need additional clarification, feel free to ask!

Here are some related questions to explore:

1. How do you find the hypotenuse of a right triangle using the Pythagorean theorem?
2. What is the sine and cosine of the calculated angle $\theta$?
3. How would you solve for the angle if you were given the hypotenuse instead of side $b$?
4. How can you verify the angle calculation using a calculator?
5. What would happen if both sides $a$ and $b$ were doubled?

Tip: The inverse trigonometric functions (arctan, arcsin, arccos) are key when working with angles in right triangles!