Certainly! Let's go through the process step-by-step to find the measure of angle \( LKJ \) using the given equation \( \tan^{-1}\left(\frac{8.9}{7.7}\right) = x \).
### Step-by-Step Solution
1. Understanding the Problem:
We are given the inverse tangent function, or arctangent, which is represented as \( \tan^{-1} \left(\frac{8.9}{7.7}\right) \). This function will provide us with the angle \( x \) whose tangent is \( \frac{8.9}{7.7} \).
2. Calculating the Tangent Ratio:
The expression inside the arctangent function is \( \frac{8.9}{7.7} \). This is the ratio of the opposite side to the adjacent side of a right-angled triangle, interpreted as the tangent of angle \( x \).
3. Applying the arctan Function:
To find \( x \), we need to determine the angle whose tangent ratio is \( \frac{8.9}{7.7} \). This means we use the arctangent function:
[tex]\[ x = \tan^{-1} \left(\frac{8.9}{7.7}\right) \][/tex]
4. Finding the Angle in Radians:
The inverse tangent of \( \frac{8.9}{7.7} \) gives us the angle in radians. From the previous calculations, the angle \( x \) in radians is:
[tex]\[ x \approx 0.857561792357106 \text{ radians} \][/tex]
5. Converting Radians to Degrees:
Often, angles are reported in degrees. To convert radians to degrees, we use the conversion factor \( 180^\circ / \pi \). Therefore:
[tex]\[ \text{Angle in degrees} = x \times \left(\frac{180^\circ}{\pi}\right) \][/tex]
6. Result in Degrees:
Performing the conversion gives us:
[tex]\[ x \approx 49.13467137373643^\circ \][/tex]
### Final Answer:
- The measure of angle \( LKJ \) is approximately \( 0.857561792357106 \) radians or \( 49.13467137373643 \) degrees.
Thus, using the arctangent function, we can find that [tex]\(\tan^{-1}\left(\frac{8.9}{7.7}\right) = 0.857561792357106\)[/tex] radians or approximately [tex]\( 49.13467137373643^\circ \)[/tex].
