To determine in which triangle the value of [tex]\( x \)[/tex] equals [tex]\( \cos^{-1}\left(\frac{4.3}{6.7} \right) \)[/tex], let's first break down the problem step-by-step:
1. Understanding the Question:
- We need to find the triangle where the angle [tex]\( x \)[/tex] has a cosine value of [tex]\( \frac{4.3}{6.7} \)[/tex].
- This involves calculating the arccosine (inverse cosine) of the fraction [tex]\( \frac{4.3}{6.7} \)[/tex].
2. Calculating the Fraction:
- The fraction [tex]\( \frac{4.3}{6.7} \approx 0.6417910447761194 \)[/tex].
3. Calculating the Angle:
- The angle [tex]\( x \)[/tex] is calculated by taking the arccosine (inverse cosine) of [tex]\( 0.6417910447761194 \)[/tex].
- This means [tex]\( x = \cos^{-1}(0.6417910447761194) \approx 0.8739648401891128 \)[/tex] radians.
- Convert this value to degrees if necessary, but since the answer is not specified in degrees or radians, we'll keep it in radians.
4. Determining the Triangle:
- To verify which triangle corresponds to this angle, we need to identify a triangle with a specific configuration.
- Usually, this would involve determining the side lengths (adjacent, opposite, and hypotenuse) to match the given cosine value.
Given the calculated cosine value (approximately 0.6418) and the resultant angle [tex]\( x \)[/tex] (approximately 0.874 radians or around 50.08 degrees), we seek a triangle where:
[tex]\[ \cos(x) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} = 0.6417910447761194 \][/tex]
Here's the step-by-step process for the comparison (assuming illustrations of triangles are available):
- Step 1: Look at each triangle's angle measures and side lengths.
- Step 2: Identify which triangle has an angle close to 0.874 radians or 50.08 degrees.
- Step 3: Verify if the cosine of the identified angle aligns with the fraction provided.
- Step 4: Check if for the identified angle, [tex]\( \cos(\text{Angle})\)[/tex] ratio of adjacent side to hypotenuse is approximately 0.6418.
Thus, we identify the triangle matching:
- It has one of its angles as [tex]\( 0.874 \)[/tex] radians (or close to 50 degrees).
- The ratio [tex]\( \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \approx 0.6418 \)[/tex].
Once you identify such a triangle, you will have the correct one satisfying the given condition.
