Sure, let's analyze the calculation step-by-step to determine the value of [tex]\( x \)[/tex].
We are given:
[tex]\[ x = \cos^{-1}\left(\frac{4.3}{6.7}\right) \][/tex]
This expression means [tex]\( x \)[/tex] is the angle whose cosine is [tex]\( \frac{4.3}{6.7} \)[/tex].
To understand this better, let's describe the process to find [tex]\( x \)[/tex] in detail without doing any calculations:
1. Identify the Ratio:
[tex]\[ \cos(x) = \frac{4.3}{6.7} \][/tex]
Here, [tex]\(\frac{4.3}{6.7}\)[/tex] is the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.
2. Calculate Cosine Inverse:
We apply the inverse cosine function to the ratio:
[tex]\[ x = \cos^{-1}\left(\frac{4.3}{6.7}\right) \][/tex]
This means [tex]\( x \)[/tex] is the angle whose cosine equals the given ratio.
3. Result in Radians and Degrees:
The calculated angle [tex]\( x \)[/tex] can be represented in both radians and degrees.
From the given information, the values are:
- [tex]\( x \approx 0.874 \)[/tex] radians
- [tex]\( x \approx 50.074 \)[/tex] degrees
So, the angle [tex]\( x \approx 50.074 \)[/tex] degrees in the context of the problem.
To see this in a triangle, consider:
- A right-angled triangle where one of the angles is approximately [tex]\( 50.074^\circ \)[/tex].
- The lengths of the triangle sides satisfy:
[tex]\[
\cos(50.074^\circ) = \frac{Adjacent}{Hypotenuse} = \frac{4.3}{6.7}
\][/tex]
Therefore, [tex]\( x \)[/tex] is the angle in a right triangle where the ratio of the adjacent side to the hypotenuse is [tex]\( \frac{4.3}{6.7} \)[/tex].
