To determine the measure of angle [tex]\( \angle LKJ \)[/tex] using the given equation [tex]\( \tan^{-1}\left(\frac{8.9}{7.7}\right) = x \)[/tex], follow these steps:
1. Understand the equation:
The equation [tex]\( \tan^{-1}\left(\frac{8.9}{7.7}\right) = x \)[/tex] involves the arctangent function, which is the inverse of the tangent function. This is used to find the angle [tex]\( x \)[/tex] whose tangent is [tex]\( \frac{8.9}{7.7} \)[/tex].
2. Calculate the tangent ratio:
The fraction [tex]\( \frac{8.9}{7.7} \)[/tex] represents the ratio of the lengths of the opposite side to the adjacent side of a right triangle.
3. Find the angle in radians:
Using the arctangent function, we can calculate the angle [tex]\( x \)[/tex] in radians:
[tex]\[
x = \tan^{-1}\left(\frac{8.9}{7.7}\right)
\][/tex]
4. Convert radians to degrees:
To convert the angle from radians to degrees, we use the fact that [tex]\( \pi \)[/tex] radians is equivalent to [tex]\( 180^\circ \)[/tex]. Therefore, the conversion factor is [tex]\( \frac{180}{\pi} \)[/tex].
5. Compute the angle in degrees:
The computed angle in degrees is approximately:
[tex]\[
x \approx 49.13467137373643^\circ
\][/tex]
6. Round to the nearest whole degree:
Finally, rounding [tex]\( 49.13467137373643^\circ \)[/tex] to the nearest whole degree gives:
[tex]\[
\boxed{49^\circ}
\][/tex]
Therefore, the measure of angle [tex]\( \angle LKJ \)[/tex] is [tex]\( 49^\circ \)[/tex], and the correct option among the provided choices is [tex]\( \boxed{49^\circ} \)[/tex].
