The triangle in the image is a right triangle, and the unknown angle $\theta$ can be determined using trigonometric ratios. Here's the information provided:

• Side opposite to $\theta$: $24$ (side $AC$).
• Side adjacent to $\theta$: $7$ (side $BC$).
• Hypotenuse: not given explicitly but can be calculated if necessary.

We can use the tangent function to find $\theta$:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{24}{7}.$

Now, calculate $\theta$:

$\theta = \tan^{-1}\left(\frac{24}{7}\right).$import math

Values of opposite and adjacent sides

opposite = 24 adjacent = 7

Calculating the angle in degrees

theta = math.degrees(math.atan(opposite / adjacent)) theta_rounded = round(theta) # Rounding to the nearest degree theta_rounded7474successThe unknown angle $\theta$ is approximately $74^\circ$ when rounded to the nearest degree.

Would you like to see the detailed steps or calculations? Here are five related questions for further exploration:

1. How can the hypotenuse of the triangle be calculated?
2. What are the other trigonometric functions (sine, cosine) of the angle $\theta$?
3. How can you confirm the solution using the Pythagorean theorem?
4. What happens to the angle if the sides are scaled proportionally?
5. How can this concept be applied in solving real-world problems?

Tip: Always double-check your calculator is set to the correct mode (degrees or radians) when solving trigonometric problems.