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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!

Find the angle θ in the given right triangle where a = 6 and b = 12. (Round your answer to one decimal place.)

Solve the problem of finding angle θ in a right triangle where the opposite side a = 6 and adjacent side b = 12. Learn to use trigonometric ratios and inverse functions to find the angle, with step-by-step guidance. Suitable for students in Grades 9-11.

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