The problem asks which trigonometric inverse function is needed to find the missing angle $\theta$ in a right triangle. Let's analyze the triangle.

Given:

• Adjacent side: $29$ (relative to the angle $\theta$)
• Opposite side: $14$
• The missing angle: $\theta$

Approach:

To determine which inverse trigonometric function to use:

1. The tangent function is defined as: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ Therefore: $\theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right) = \tan^{-1}\left(\frac{14}{29}\right)$

Thus, the correct choice is $\tan^{-1}$.

Would you like me to calculate the angle or explain further?

Related Questions:

1. What are the definitions of sine, cosine, and tangent functions?
2. How can inverse trigonometric functions be used in real-world applications?
3. What is the difference between $\tan^{-1}$ and $\cot^{-1}$?
4. How do we confirm that this triangle adheres to the Pythagorean theorem?
5. How can calculators or software simplify solving triangles like this?

Tip:

Always match the known sides of a triangle (opposite, adjacent, hypotenuse) to the correct trigonometric function for efficient problem-solving!