To determine the unknown angle $\theta$ in the given right triangle, we can use trigonometric relationships. Based on the given dimensions:

• The adjacent side to $\theta$ is $7$.
• The opposite side to $\theta$ is $24$.

We can use the tangent function:

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

1. Compute $\tan^{-1}$ to find $\theta$:

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

Let me calculate this.import math

Given values

opposite = 24 adjacent = 7

Calculate theta using arctan

theta_radians = math.atan(opposite / adjacent) theta_degrees = math.degrees(theta_radians) theta_degrees_rounded = round(theta_degrees)

theta_degrees_rounded7474successThe measurement of the unknown angle $\theta$ is approximately $74^\circ$, rounded to the nearest degree.

Would you like a detailed breakdown of the trigonometric steps or further assistance? Here are some related questions to explore:

1. What are the values of the other trigonometric ratios for this triangle?
2. How would the angle change if the side lengths were doubled?
3. Can you calculate the area of the triangle using these dimensions?
4. What would the hypotenuse of this triangle be?
5. How does the tangent function change for angles beyond $90^\circ$?

Tip: Always double-check the side labels (opposite, adjacent, hypotenuse) when using trigonometric functions. Mislabeling can lead to incorrect results.