To solve this problem, we need to determine the angle [tex]\( x \)[/tex] in a triangle where [tex]\( x = \cos^{-1}\left(\frac{4.3}{6.7}\right) \)[/tex].
Let's break down the problem step-by-step:
1. Understanding the cosine inverse (arccos) function:
- The function [tex]\(\cos^{-1}(y)\)[/tex] gives the angle [tex]\( x \)[/tex] whose cosine is [tex]\( y \)[/tex].
- [tex]\( x = \cos^{-1}\left(\frac{4.3}{6.7}\right) \)[/tex] means [tex]\( \cos(x) = \frac{4.3}{6.7} \)[/tex].
2. Interpreting the ratio:
- The expression [tex]\(\cos(x) = \frac{4.3}{6.7}\)[/tex] can be interpreted geometrically: In a right triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse.
3. Evaluating the specific cosine value:
- Given [tex]\( \cos(x) = \frac{4.3}{6.7} \)[/tex], the specific angle [tex]\( x \)[/tex] is determined by finding the arccosine of that ratio. This evaluated angle is approximately [tex]\( x = 0.8739648401891128 \)[/tex] radians.
4. Identifying the triangle:
- To determine in which triangle this angle appears, imagine a right triangle where:
- The length of the side adjacent to [tex]\( x \)[/tex] (let's call it [tex]\( a \)[/tex]) is 4.3 units.
- The length of the hypotenuse (let's call it [tex]\( c \)[/tex]) is 6.7 units.
- Using these lengths, the angle [tex]\( x \)[/tex] we calculated is [tex]\( \cos^{-1}\left(\frac{4.3}{6.7}\right) = 0.8739648401891128 \)[/tex] radians.
In conclusion, look for a triangle with sides corresponding to these specific lengths, where the adjacent side to angle [tex]\( x \)[/tex] is 4.3 units and the hypotenuse is 6.7 units. The angle [tex]\( x \)[/tex] in that particular triangle will be [tex]\( \cos^{-1}\left(\frac{4.3}{6.7}\right) \)[/tex], which is approximately 0.8739648401891128 radians.
