To solve for the angle $\theta$ in the right triangle shown in the image, we can apply trigonometric relationships.

The right triangle has:

• The opposite side of angle $\theta$ as 2,
• The adjacent side as 1, and
• The hypotenuse as 5.

Using the tangent function, which relates the opposite and adjacent sides:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{2}{1} = 2$

Now, to find $\theta$, we can take the arctangent (inverse tangent) of 2:

$\theta = \tan^{-1}(2)$

Let me calculate that for you.The angle $\theta$ is approximately $63.43^\circ$.

Would you like further details on the solution? Here are some related questions:

1. How can we verify this result using the sine or cosine functions?
2. How would you calculate the other angle in the triangle?
3. What would happen if the sides of the triangle were different?
4. How does the Pythagorean theorem apply to this problem?
5. Can the law of sines or cosines be used to solve this triangle?

Tip: In trigonometry, the tangent function is particularly useful for right triangles when only the opposite and adjacent sides are known.