To find the angle $\theta$ in a right triangle where the side opposite $\theta$ is $6$ and the side adjacent to $\theta$ is $24$, you can use the tangent function:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{6}{24}$

Step 1: Calculate the Tangent Ratio

$\tan(\theta) = \frac{6}{24} = 0.25$

Step 2: Find $\theta$ Using the Inverse Tangent

Now, take the inverse tangent (arctan) of $0.25$ to find $\theta$: $\theta = \arctan(0.25)$

Using a calculator: $\theta \approx 14.0^\circ$

Answer:

$\theta \approx 14.0^\circ$

Let me know if you'd like more details or have questions. Here are related problems you might find useful:

1. How would you find $\theta$ if you were given the hypotenuse and the opposite side?
2. What is the angle $\theta$ if the sides are scaled by a factor?
3. How would you find the adjacent side if given $\theta$ and the opposite side?
4. What happens to $\theta$ if the opposite side is doubled?
5. Can you solve for $\theta$ using cosine if both the adjacent and hypotenuse are known?

Tip: When using trigonometric ratios, remember which sides correspond to opposite, adjacent, and hypotenuse relative to the angle you’re calculating.