To solve for the angle opposite the side labeled $h$ in the given triangle, we'll analyze the triangle using trigonometric principles.

Analyzing the Triangle:

1. Given Information:

   • The smaller triangle has one angle of $22^\circ$, the opposite side is $h$, and the adjacent side is $2h$.
   • The right triangle has a right angle.

2. Finding the Angle:

   • Let's denote the angle we need to find as $\theta$, which is opposite the side $h$.

   In the larger right triangle, the tangent of the angle $\theta$ can be defined as: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{2h} = \frac{1}{2}$

3. Using the Tangent Inverse:

   • To find the angle $\theta$, take the inverse tangent: $\theta = \tan^{-1}\left(\frac{1}{2}\right)$

Calculation:

$\theta \approx \tan^{-1}\left(0.5\right) \approx 26.565^\circ$

Thus, the angle opposite the side labeled $h$ is approximately 26.565°.

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Would you like further details or have any questions? Here are some related topics you might find interesting:

1. How to calculate angles in non-right triangles?
2. What is the relationship between the sides and angles in a right triangle?
3. How do inverse trigonometric functions work?
4. What is the sine rule and when is it used?
5. How to solve triangles using the cosine rule?
6. What are complementary and supplementary angles?
7. How do you solve a triangle when all sides are known?
8. How can trigonometric identities help simplify complex calculations?

Tip: Understanding the basic trigonometric functions (sine, cosine, tangent) is key to solving various problems involving angles and distances in triangles.