The problem asks us to evaluate the trigonometric function $\tan^{-1}(0.6877)$ to determine the quadrant of the angle.

Step-by-Step Solution

1. Determine the Angle: We have $\tan \theta = 0.6877$. Since $\tan \theta$ is positive, the angle $\theta$ could be in either the first quadrant or the third quadrant, as these are the quadrants where the tangent function is positive.

2. Range of Values: Typically, inverse tangent functions (or arctangent) return an angle in the range of $-\frac{\pi}{2}$ to $\frac{\pi}{2}$ (which corresponds to the first and fourth quadrants) on calculators and in standard trigonometric functions. However, since $\tan 0.6877$ is positive and the range of the principal value for arctan is the first quadrant, this implies that the angle is in the first quadrant.

3. Final Answer:

   • Since $0.6877$ is within the range associated with the first quadrant, we can conclude:
     • The angle is in Quadrant 1.

If you have any questions or would like additional details, feel free to ask!

Further Questions for Practice

1. What is the arctangent of a value that would place the angle in the third quadrant?
2. If $\tan \theta = -0.6877$, what quadrant would $\theta$ be in?
3. How does the principal range of the inverse tangent function affect which quadrants angles are placed in?
4. Can you calculate $\arctan(0.6877)$ in radians or degrees?
5. What are the ranges of the tangent function in each quadrant?

Tip: Remember that the tangent function is positive in the first and third quadrants. This helps determine the quadrant based on the sign of $\tan \theta$.